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\subsection{Structure of trees}\label{Section_structure_of_trees}
Here, we gather lemmas about the structure of trees. Most of these lemmas say something about the leaves and bare paths of a tree. It is easy to see that a tree with few leaves must have many bare paths. The most common version of this is the following well known lemma.
\begin{lemma}[\cite{montgomery2018spanning}]\label{split} For any integers $n,k>2$, a tree with~$n$ vertices either has at least $n/4k$ leaves or a collection of at least $n/4k$ vertex disjoint bare paths, each with length $k$.
\epsilonilonnd{lemma}
As a corollary of this lemma we show that every tree either has many bare paths, many non-neighbouring leaves, or many large stars. This lemma underpins the basic case division for this paper. The rest of the proofs focus on finding rainbow copies of the three types of trees.
\begin{lemma}[Case division]\label{Lemma_case_division}
Let $1\gg \delta \gg n^{-1}$. Every $n$-vertex tree satisfies one of the following:
\begin{enumerate}[label= \Alph{enumi}]
\item There are at least $\delta n/800$ vertex-disjoint bare paths with length at least $\delta^{-1}$.
\item There are at least $\delta^6 n$ non-neighbouring leaves.
\item Removing leaves next to vertices adjacent to at least $\delta^{-4}$ leaves gives a tree with at most
$n/100$ vertices.
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
\begin{proof}
Take $T$ and remove leaves around any vertex adjacent to at least $\delta^{-4}$ leaves, and call the resulting tree $T'$. If $T'$ has at most $n/100$ vertices then we are in Case C. Assume then, that $T'$ has at least $n/100$ vertices.
By Lemma~\ref{split} applied with $n'=|T'|$ and $k=\lceil \delta^{-1}\rceil$, the tree $T'$ either has at least $\delta n/600$ vertex disjoint bare paths with length at least $\delta^{-1}$ or at least $\delta n/600$ leaves. In the first case, as vertices were deleted next to at most $\delta^4 n$ vertices to get $T'$ from $T$, $T$ has at least $\delta n/600-\delta^4n\geq \delta n/800$ vertex disjoint bare paths with length at least $\delta^{-1}$, so we are in Case B.
In the second case, if there are at least $\delta n/1200$ leaves of $T'$ which are also leaves of $T$, then, as there are at most $\delta^{-4}$ of these leaves around each vertex in $T'$, $T$ has at least $(\delta n/1200)/\delta^{-4}\geq \delta^{6}n$ non-neighbouring leaves, so we are in Case A. On the other hand, if there are not such a number of leaves of $T'$ which are leaves of $T$, then $T'$ must have at least $\delta n/1200$ leaves which are not leaves of $T$, and which therefore are adjacent to a leaf in $T$. Thus, $T$ has at least $\delta n/1200\geq \delta^{6}n$ non-neighbouring leaves, and we are also in Case A.
\epsilonilonnd{proof}
We say a set of subtrees $T_1,\ldots, T_\epsilonilonll\subseteq T$ divides a tree $T$ if $E(T_1)\cup\ldots \cup E(T_\epsilonilonll)$ is a partition of $E(T)$. We use the following lemma.
\begin{lemma}[\cite{montgomery2018spanning}]\label{littletree} Let $n,m\in \mathbb N$ satisfy $1\leq m\leq n/3$. Given any tree~$T$ with~$n$ vertices and a vertex $t\in V(T)$, we can find two trees $T_1$ and $T_2$ which divide~$T$ so that $t\in V(T_1)$ and $m\leq |T_2|\leq 3m$.
\epsilonilonnd{lemma}
Iterating this, we can divide a tree into small subtrees.
\begin{lemma}\label{dividetree} Let $T$ be a tree with at least $m$ vertices, where $m\geq 2$. Then, for some $s$, there is a set of subtrees $T_1,\ldots, T_s$ which divide $T$ so that $m\leq |T_i|\leq 4m$ for each $i\in [s]$.
\epsilonilonnd{lemma}
\begin{proof}
We prove this by induction on $|T|$, noting that it is trivially true if $|T|\leq 4m$.
Suppose then $|T|>4m$ and the statement is true for all trees with fewer than $|T|$ vertices and at least $m$ vertices. By Lemma~\ref{littletree}, we can find two trees $T_1$ and $S$ which divide $T$ so that $m\leq |T_1|\leq 3m$. As $|T|>4m$, we have $m<|S|<|T|$, so there must be a set of subtrees $T_2,\ldots,T_s$, for some $s$, which divide $S$ so that $m\leq |T_i|\leq 4m$ for each $2\leq i\leq s$.
The subtrees $T_1,\ldots,T_s$ then divide $T$, with $m\leq |T_i|\leq 4m$ for each $i\in [s]$.
\epsilonilonnd{proof}
For embedding trees in Cases A and B, we will need a finer understanding of the structure of trees. In fact, every tree can be built up from a small tree by successively adding leaves, bare paths, and stars, as follows.
\begin{lemma}[\cite{MPS}]\label{Lemma_decomp} Given integers $d$ and $n$, $\mu>0$ and a tree $T$ with at most $n$ vertices, there are integers $\epsilonilonll\leq 10^4 d\mu^{-2}$ and $j\in\{2,\ldots,\epsilonilonll\}$ and a sequence of subgraphs $T_0\subseteq T_1\subseteq \ldots \subseteq T_\epsilonilonll=T$ such that
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item for each $i\in [\epsilonilonll]\setminus \{1,j\}$, $T_{i}$ is formed from $T_{i-1}$ by adding non-neighbouring leaves,\label{cond1}
\item $T_j$ is formed from $T_{j-1}$ by adding at most $\mu n$ vertex-disjoint bare paths with length $3$,\label{cond2}
\item $T_1$ is formed from $T_0$ by adding vertex-disjoint stars with at least $d$ leaves each, and\label{cond3}
\item $|T_0|\leq 2\mu n$.\label{cond4}
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
The following variation will be more convenient to use here. It shows that an arbitrary tree $T$ can be built out of a preselected, small subtree $T_1$ by a sequence of operations. It is important to control the starting tree as it allows us to choose which part of the tree will form our absorbing structure.
For forests $T'\subseteqeq T$, we say that $T$ is obtained from $T'$ by \epsilonilonmph{adding a matching of leaves} if all the vertices in $V(T)\setminus V(T')$ are non-neighbouring leaves in $T$.
\begin{lemma}[Tree splitting]\label{Lemma_tree_splitting}
Let $1\geq d^{-1} \gg n^{-1}$.
Let $T$ be a tree with $|T|=n$ and $U\subseteqeq V(T)$ with $|U|\geq n/d^3$.
Then, there are forests $T_1^{\mathrm{small}}\subseteqeq T_2^{\mathrm{stars}}\subseteqeq T_3^{\mathrm{match}}\subseteqeq T_4^{\mathrm{paths}}\subseteqeq T_5^{\mathrm{match}}=T$ satisfying the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $|T_{1}^{\mathrm{small}}|\leq n/d$ and $|U\cap T_1^{small}|\geq n/d^6$.\label{rush1}
\item $T_2^{\mathrm{stars}}$ is formed from $T_{1}^{\mathrm{small}}$ by adding vertex-disjoint stars of size at least $d$.\label{rush2}
\item $T_3^{\mathrm{match}}$ is formed from $T_2^{\mathrm{stars}}$ by adding a sequence of $d^8$ matchings of leaves.\label{rush3}
\item $T_4^{\mathrm{paths}}$ is formed from $T_3^{\mathrm{match}}$ by adding at most $n/d$ vertex-disjoint paths of length $3$.\label{rush4}
\item $T_5^{\mathrm{match}}$ is formed from $T_4^{\mathrm{paths}}$ by adding a sequence of $d^8$ matchings of leaves.\label{rush5}
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
\begin{proof}
First, we claim that there is a subtree $T_0$ of order $\leq n/2d^2$ containing at least $n/d^6$ vertices of $U$. To find this, use Lemma~\ref{dividetree} to find $s\leq 32d^2$ subtrees $T_1,\ldots, T_{s}$ which divide $T$ so that $n/16d^2\leq |T_i|\leq n/2d^2$ for each $i\in [s]$. As each vertex in $U$ must appear in some tree $T_i$, there must be some tree $T_k$ which contains at least $|U|/32d^2\geq n/d^6$ vertices in $U$, as required.
Let $T'$ be the $n$-vertex tree formed from $T$ by contracting $T_k$ into a single vertex $v_0$ and adding $e(T_k)$ new leaves at $v_0$ (called ``dummy'' leaves).
Notice that Lemma~\ref{Lemma_decomp} applies to $T'$ with $d=d, \mu= d^{-3}, n=n$ which gives a sequence of forests $T_0', \dots, T_{\epsilonilonll}'$ for $\epsilonilonll\leq 10^4d^7\leq d^8$.
Notice that $v_0\in T_0'$. Indeed, by construction, every vertex which is not in $T_0'$ can have in $T_{\epsilonilonll}'=T'$ at most $\epsilonilonll$ leaves. Since $v_0$ has $\geq e(T_k)>\epsilonilonll$ leaves in $T'$, it must be in $T'_0$.
For each $i=0, \dots, {\epsilonilonll}$, let $T_i''$ be $T_i'$ with $T_k$ uncontracted and any dummy leaves of $v_0$ deleted.
Let $T_2^{\mathrm{stars}}=T_1''$, $T_3^{\mathrm{match}}=T_{j-1}''$, $T_4^{\mathrm{paths}}=T_j''$, $T_5^{\mathrm{match}}=T_{\epsilonilonll}''$.
Let $T_1^{\mathrm{small}}$ be $T_0''$ together with the $T_1''$-leaves of any $v\in T_k$ for which $|N_{T_1''}(v)\setminus N_{T_0''}(v)|<d$. We do this because when we uncontract $T_k$ the leaves which were attached to $v_0$ in $T_0'$ are now attached to vertices of $T_k$.
If they form a star of size less than $d$ we cannot add them when we form $T_2^{\mathrm{stars}}$ without violating \ref{rush2} so we add them already when we form $T_1^{\mathrm{small}}$.
Since we are adding at most $d$ leaves for every vertex of $T_k$, we have $|T_1^{\mathrm{small}}|\leq d|T_k|+|T_0'|\leq n/d$ so \ref{rush1} holds. This ensures that at least $d$ leaves are added to vertices of $T_1^{\mathrm{small}}$ to form $T_2^{\mathrm{stars}}$ so \ref{rush2} holds. The remaining conditions \ref{rush3} -- \ref{rush5} are immediate from the application of Lemma~\ref{Lemma_decomp}.
\epsilonilonnd{proof}
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\subsection{Pseudorandom properties of random sets of vertices and colours}\label{Section_pseudorandomness}
Suppose that $K_{2n+1}$ is $2$-factorized. Choose a $p$-random set of vertices $V\subseteqeq V(K_{2n+1})$ and a $q$-random set of colours $C\subseteqeq C(K_{2n+1})$. What can be said about the subgraph $K_{2n+1}[V,C]$ consisting of edges within the set $V$ with colour in $C$? What ``pseudorandomness'' properties is this subgraph likely to have? In this section, we gather lemmas giving various such properties.
The setting of the lemmas is quite varied, as are the properties they give. For example, sometimes the sets $V$ and $C$ are chosen independently, while sometimes they are allowed to depend on each other arbitrarily. We split these lemmas into three groups based on the three principal settings.
\subsubsectionme{Dependent vertex/colour sets}
In this setting, our colour set $C\subseteqeq C(K_{2n+1})$ is $p$-random, and the vertex set is $V(K_{2n+1})$ (i.e., it is 1-random). Our pseudorandomness condition is that the number of edges between any two sizeable disjoint vertex sets is close to the expected number. Though a lemma of this kind was first proved in~\cite{alon2016random}, the precise pseudorandomness condition we will use here is in the following version from~\cite{MPS}. A colouring is \epsilonilonmph{locally $k$-bounded} if every vertex is adjacent to at most $k$ edges of each colour.
\begin{lemma}[\cite{MPS}]\label{MPScolour}\label{Lemma_MPS_boundrandcolour} Let $k\in\mathbb N$ be constant and let $\epsilonilonpsilon, p\geq n^{-1/100}$. Let $K_n$ have a {locally} $k$-bounded colouring and suppose $G$ is a subgraph of $K_n$ chosen by including the edges of each colour independently at random with probability $p$. Then, with probability $1-o(n^{-1})$, for any disjoint sets $A,B\subseteq V(G)$, with $|A|,|B|\geq n^{3/4}$,
\[
\big|e_G(A,B)-p|A||B|\big|\leq \epsilonilonpsilon p|A||B|.
\]
\epsilonilonnd{lemma}
If $V\subseteqeq V(K_{2n+1})$ is a $p$-random set of vertices, then the edges going from $V$ to $V(K_{2n+1})\setminus V$ are typically pseudorandomly coloured. The lemma below is a version of this. Here ``pseudorandomly coloured'' means that most colours have at most a little more than the expected number of colours leaving $V$.
\begin{lemma}[\cite{MPS}]\label{MPSvertex}\label{Corollary_MPS_randsetcor}
Let $k$ be constant and $\epsilonilonpsilon,p\geq n^{-1/10^3}$. Let $K_n$ have a {locally} $k$-bounded colouring and let $V$ be a $p$-random subset of $V(K_n)$. Then, with probability $1-o(n^{-1})$, for each $A\subseteq V(K_n)\setminus V$ with $|A|\geq n^{1/4}$, for all but at most $\epsilonilonpsilon n$ colours there are at most $(1+\epsilonilonpsilon)pk|A|$ edges of that colour between $V$ and $A$.
\epsilonilonnd{lemma}
A random vertex set $V$ likely has the property from Lemma~\ref{MPSvertex}. A random colour set $C$ likely has the property from Lemma~\ref{MPScolour}. If we combine these two properties, we can get a property involving $C$ and $V$ that is likely to hold. Importantly, this will be true even if $C$ and $V$ are not independent of each other. Doing this, we get the following lemma.
\begin{lemma}[Nearly-regular subgraphs]\label{Lemma_nearly_regular_subgraph}
Let $p, \gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$, and $C\subseteqeq C(K_{2n+1})$ with $V$ $p/2$-random and $C$ $p$-random (possibly depending on each other). The following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|= pn$, there are subsets $U'\subseteqeq U, V'\subseteqeq V, C'\subseteqeq C$ with $|U'|=|V'|= (1\pm\gamma)|U|$
so that $G=K_{2n+1}[U',V',C']$ is globally $(1+\gamma)p^2n$-bounded, and every vertex $v\in V(G)$ has $d_{G}(v)=(1\pm\gamma)p^2n$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ and $\epsilonilonpsilon$ so that $p, \gamma\gg \alpha\gg \epsilonilonpsilon\gg n^{-1}$. With high probability, by Lemma~\ref{MPScolour}, Lemma~\ref{MPSvertex} (with $k=2$, $n'=2n+1$, and $p'=p/2$) and Chernoff's bound, we can assume the following occur simultaneously.
\begin{enumerate}[label = (\roman{enumi})]
\item \label{prop1} For any disjoint $A,B\subseteq V(K_{2n+1})$ with $|A|,|B|\geq (2n+1)^{3/4}$, $|E_C(A,B)|=(1\pm \epsilonilonps)p|A||B|$.
\item \label{prop2} For any $A\subseteqeq V(K_{2n+1})\setminus V$ with $|A|\geq n^{1/4}$, for all but at most $\epsilonilonpsilon n$ colours there are at most $(1+\epsilonilonpsilon)p|A|$ edges of that colour between $A$ and $V$.
\item \label{prop3} $|V|=(1\pm \epsilonilonps)pn$.
\epsilonilonnd{enumerate}
Let $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|= pn$ be arbitrary. Let $\hat C\subseteqeq C$ be the subset of colours $c\in C$ with $|E_c(U,V)|> (1+ \epsilonilonpsilon)p|U|$. Note that, from \ref{prop2}, we have $|\hat C|\leq \epsilonilonps n$.
Let $\hat U^+\subseteqeq U$ and $\hat V^+\subseteqeq V$ be subsets of vertices $v$ with $d_{K_{2n+1}[U,V,C]}(v)> (1+ \alpha)p^2 n$, and let $\hat U^-\subseteqeq U$ and $\hat V^-\subseteqeq V$ be subsets of vertices $v$ with $d_{K_{2n+1}[U,V,C]}(v)< (1- \alpha)p^2 n$.
Now, $|E_C(U^+,V)|> |U^+|(1+\alpha)p^2 n\geq (1+\epsilonilonps)p|U^+||V|$ by the definition of $U^+$ and \ref{prop3}. Therefore, as, by \ref{prop3}, $|V|\geq (1-\epsilonilonps)pn>(2n+1)^{3/4}$, from \ref{prop1} we must have $|U^+|\leq (2n+1)^{3/4}\leq \epsilonilonps n$. Similarly, we have $|U^-|,|V^+|,|V^-|\leq \epsilonilonps n$.
Let $\hat U=U^+\cup U^-$ and Let $\hat V=V^+\cup V^-$. We have $|U\setminus \hat U|, |V\setminus \hat V| \geq (1\pm \epsilonilonpsilon)pn\pm 2\epsilonilonpsilon n$. Therefore, we can choose subsets $U'\subseteqeq U\setminus \hat U$ and $V'\subseteqeq V\setminus \hat V$ with $|U'|=|V'|= pn-3\epsilonilonpsilon n=(1\pm \gamma)pn$. Note that, by \ref{prop3} and as $|U|=pn$, $|U\setminus U'|,|V\setminus V'|\leq 4\epsilonilonpsilon n$.
Let $C'=C\setminus \hat C$ and set $G=K_{2n+1}[U',V',C']$.
For each $v\in U'\cup V'$ we have $d_G(v)= (1\pm \alpha)p^2 n \pm 2|\hat C|\pm |U\setminus U'|\pm |V\setminus V'|=(1\pm \gamma)p^2 n$. For each $c\in C'$ we have $|E_c(U,V)|\leq (1+ \epsilonilonpsilon)p|U|\leq (1+ \gamma)p^2n$.
\epsilonilonnd{proof}
\subsubsectionme{Deterministic colour sets and random vertex sets}
The following lemma bounds the number of edges each colour typically has within a random vertex set.
\begin{lemma}[Colours inside random sets]\label{Lemma_number_of_colours_inside_random_set}
Let $p,\gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p$-random. With high probability, every colour has $(1\pm \gamma)2p^2n$ edges inside $V$.
\epsilonilonnd{lemma}
\begin{proof}
For an edge $e\in E(K_{2n+1})$ we have $\mathbb{P}(e \in E(K_n[V]))=p^2$.
By linearity of expectation, for any colour $c\in C(K_{2n+1})$, we have $\mathbb{E}(|E_c(V)|)=p^2(2n+1)$. Note that $|E_c(V)|$ is $2$-Lipschitz and affected by $\leq 2n+1$ coordinates.
By Azuma's inequality, we have $\mathbb{P}(|E_c(V)|\neq (1\pm \gamma)2p^2n )\leq e^{-\gamma^2p^4n/100}= o(n^{-1})$.
The result follows by a union bound over all the colours.
\epsilonilonnd{proof}
Recall that for any two sets $U,V\subseteqeq V(G)$ inside a coloured graph $G$, we say that the pair $(U,V)$ is \epsilonilonmph{$k$-replete} if every colour of $G$ occurs at least $k$ times between $U$ and $V$.
We will use the following auxiliary lemma about how this property is inherited by random subsets.
\begin{lemma}[Repletion between random sets]\label{Lemma_inheritence_of_lower_boundedness_random}
Let $q, p\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Suppose that $A,B\subseteqeq V(K_{2n+1})$ are disjoint randomized sets with the pair $(A,B)$ $pn$-replete with high probability.
Let $V\subseteqeq V(K_{2n+1})$ be $q$-random and independent of $A,B$.
Then with high probability the pair $(A, B\cap V)$ is $(qpn/2)$-replete.
\epsilonilonnd{lemma}
\begin{proof}
Fix some choice $A'$ of $A$ and $B'$ of $B$ for which the pair $(A',B')$ is $pn$-replete. As $V$ is independent of $A,B$, for each edge $e$ between $A$ and $B$ we have $\mathbb{P}(e\cap B\cap V\neq \epsilonilonmptyset|A=A', B=B')=q$.
Therefore, for any colour $c$, conditional on ``$A=A', B=B'$'', we have $\mathbb{E}(|E_c(A,B\cap V)|)=q|E_c(A,B)|\geq qpn$. Note that $|E_c(A,B\cap V)|$ is $2$-Lipschitz and affected by $\leq 2n+1$ coordinates.
By Azuma's inequality, we have $\mathbb{P}(|E_c(A,B\cup V)|< qpn/2| A=A', B=B')\leq e^{-q^2p^2n/100}= o(1)$. Thus, with probability $1-o(1)$, conditioned on $A=A', B=B'$ we have that $(A,B\cap V)$ is $(qn/2)$-replete.
This was all under the assumption that $A=A'$ and $B=B'$.
Therefore using that $(A,B)$ is $pn$-replete with high probability, we have
\begin{align*}
\mathbb{P}(\text{$(A,B\cap V)$ is $(qn/2)$-replete})&\geq \sum_{\substack{(A',B') \\ \text{ $pn$-replete}}}\mathbb{P}(\text{$(A,B\cap V)$ is $(qn/2)$-replete}|A=A', B=B')\cdot\mathbb{P}(A=A', B=B')\\
&\geq \sum_{\substack{(A',B') \\ \text{ $pn$-replete}}} (1-o(1))\cdot \mathbb{P}(A=A', B=B')=1-o(1).\qed
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here
\epsilonilonnd{align*}
\epsilonilonnd{proof}
\subsubsectionme{Independent vertex/colour sets}
The setting of the next three lemmas is the same: we independently choose a $p$-random set of vertices $V$ and a $q$-random set of colours $C$. For such a pair $V,C$ we expect all vertices of the vertices $v$ in $K_{2n+1}$ to have many $C$-edges going into $V$. Each of the following lemmas is a variation on this theme.
\begin{lemma}[Degrees into independent vertex/colour sets]\label{Lemma_high_degree_into_random_set}
Let $p, q\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p$-random, and let $C\subseteqeq C(K_{2n+1})$ be $q$-random and independent of $V$. With probability $1-o(n^{-1})$, every vertex $v\in V(K_{2n+1})$ has $|N_C(v)\cap V|\geq pq n$.
\epsilonilonnd{lemma}
\begin{proof} Let $v\in V(K_{2n+1})$.
For any vertex $x\neq v$, we have $\mathbb{P}(x\in N_C(v)\cap V)= pq$ and so $\mathbb{E}(|N_C(v)\cap V|)= 2pq n$.
Also $|N_C(v)\cap V|$ is $2$-Lipschitz and affected by $3n$ coordinates.
By Azuma's Inequality, we have that $\mathbb{P}(|N_C(v)\cap V|\leq pq n)\leq 2e^{-p^2q^{2}n/1000}= o(n^{-2})$. The result follows by taking a union bound over all $v\in V(K_{2n+1})$.
\epsilonilonnd{proof}
\begin{lemma}\label{lem:goodpairs} Let $1/n\ll \epsilonilonta \ll \mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$-random subsets which are independent. With high probability, for each distinct $u,v\in V(K_{2n+1})$, there are at least $\epsilonilonta n$ colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$.
\epsilonilonnd{lemma}
\begin{proof} Let $u,v\in V(K_{2n+1})$ be distinct, and let $X_{u,v}$ be the number of colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$. Note that $\mathbb{E} X_{u,v} \geq \mu^3 n$, $X_{u,v}$ is
$2$-Lipschitz and affected by $3n-1$ coordinates.
By Azuma's Inequality, we have that $\mathbb{P}(X_{u,v}\leq \mu^3 n/2)\leq 2e^{-\mu^6 n/1000}= o(n^{-2})$. The result follows by taking a union bound over all distinct pairs $u,v\in V(K_{2n+1})$.
\epsilonilonnd{proof}
Lemma~\ref{Lemma_high_degree_into_random_set} says that, with high probability, every vertex $v$ has many colours $c\in C$ for which there is a $c$-edge into $V$. The following lemma is a strengthening of this. It shows that, for any set $Y$ of $100$ vertices, there are many colours $c\in C$ for which \epsilonilonmph{each $v\in Y$} has a $c$-edge into $V$.
\begin{lemma}[Edges into independent vertex/colour sets]\label{Lemma_edges_into_independent_vertexcolour_sets}
Let $p\gg q\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ be $p$-random and independent.
Then, with high probability, for any set $Y$ of $100$ vertices, there are $qn$ colours $c\in C$ for which each $y\in Y$ has a $c$-neighbour in $V$.
\epsilonilonnd{lemma}
\begin{proof} Fix $Y\subseteq V(K_{2n+1})$ with $|Y|=100$.
Let $C_Y=\{c\in C: \mbox{each $y\in Y$ has a $c$-neighbour in $V$}\}$.
For any colour $c$ without edges inside $Y$, we have $\mathbb{P}(c\in C_Y)\geq p^{101}$ and so $\mathbb{E}(|C_Y|)\geq p^{101}(n-\binom{|Y|}2)\geq 2qn$. Notice that $|C_Y|$ is $100$-Lipschitz and affected by $3n+1$ coordinates.
By Azuma's inequality, we have that $\mathbb{P}(|C_Y|\leq qn)\leq e^{-q^2n/10^6}= o(n^{-100})$. The result follows by taking a union bound over all sets $Y\subseteq V(K_{2n+1})$ with $|Y|=100$.
\epsilonilonnd{proof}
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\subsection{Rainbow matchings}\label{Section_matchings}
We now gather lemmas for finding large rainbow matchings in random subsets of coloured graphs, despite dependencies between the colours and the vertices that we use. Simple greedy embedding strategies are insufficient for this, and instead we will use a variant of R\"odl's Nibble proved by the authors in~\cite{montgomery2018decompositions}.
\begin{lemma}[\cite{montgomery2018decompositions}]\label{Lemma_MPS_nearly_perfect_matching}
Suppose that we have $n, \delta, \gamma, p, \epsilonilonll$ with $1\geq \delta \gg p \gg \gamma \gg n^{-1}$ and $n\gg \epsilonilonll$.
Let $G$ be a locally $\epsilonilonll$-bounded, globally $(1+\gamma) \delta n$-bounded, coloured, balanced bipartite graph with $|G|=(1\pm \gamma)2n$ and $d_G(v)=(1\pm \gamma)\delta n$ for all $v\in V(G)$. Then $G$ has a random rainbow matching $M$ which has size $\geq (1-2p)n$ where
\begin{align}
\mathbb{P}(e\in E(M))&\geq(1- 9p)\frac{1}{\delta n}\hspace{0.5cm}\text{ for each $e\in E(G)$.} \label{Eq_Near_Matching_Edge_Probability_Lower_Bound}
\epsilonilonnd{align}
\epsilonilonnd{lemma}
We remark that in the statement of this lemma \cite{montgomery2018decompositions}, the conditions
``$|G|=(1\pm \gamma)2n$ and $d_G(v)=(1\pm \gamma)\delta n$ for all $v\in V(G)$'' are referred to collectively as ``$G$ is $(\gamma, \delta, n)$-regular''.
The following lemma is at the heart of the proofs in this paper. It shows there is typically a nearly-perfect rainbow matching using random vertex/colour sets. Moreover, it allows \epsilonilonmph{arbitrary} dependencies between the sets of vertices and colours. As mentioned before, when embedding high degree vertices such dependencies are unavoidable. Because of this, after we have embedded the high degree vertices, the remainder of the tree will be embedded using variants of this lemma.
\begin{lemma}[Nearly-perfect matchings]\label{Lemma_nearly_perfect_matching}
Let $p\in [0,1]$, $\beta \gg n^{-1}$, and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p/2$-random and let $C\subseteqeq C(K_{2n+1})$ be $p$-random (possibly depending on each other). Then, with probability $1-o(n^{-1})$, for every $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|\leq pn$, $K_{2n+1}$ has a $C$-rainbow matching of size $|U|-\beta n$ from $U$ to $V$.
\epsilonilonnd{lemma}
\begin{proof}
The lemma is vacuous when $p<\beta$, so suppose $p\geq \beta$. We will first prove the lemma in the special case when $p\leq 1-\beta$.
Choose $p\geq \beta\gg \alpha \gg \gamma\gg n^{-1}$.
With probability $1-o(n^{-1})$, $V$ and $C$ satisfy the conclusion of Lemma~\ref{Lemma_nearly_regular_subgraph} with $p, \gamma, n$.
Using Chernoff's bound and $p\leq 1-\beta$, with probability $1-o(n^{-1})$ we have $|V|\leq n$. By the union bound, both of these simultaneously occur.
Notice that it is sufficient to prove the lemma for sets $U$ with $|U|=pn$ (since any smaller set $U$ is contained in a set of this size which is disjoint from $V$ as $|V| \leq n$).
From Lemma~\ref{Lemma_nearly_regular_subgraph}, we have that, for $U$ of order $pn$, there are subsets $U'\subseteqeq U, V'\subseteqeq V, C'\subseteqeq C$ with $|U'|=|V'|= (1\pm\gamma)pn$
so that $G=K_{2n+1}[U',V',C']$ is globally $(1+\gamma)p^2n$-bounded, and every vertex $v\in V(G)$ has $d_{G}(v)=(1\pm\gamma)p^2n$. Now $G$ satisfies the assumptions of Lemma~\ref{Lemma_MPS_nearly_perfect_matching} (with $n'=pn$, $\delta =p$, $p'= \alpha$, $\gamma'=2\gamma$ and $\epsilonilonll=2$), so it has a rainbow matching of size $(1-2\alpha)pn\geq pn-\beta n$.
Now suppose that $p\geq 1-\beta$. Choose a $(1-\beta/2)p/2$-random subset $V'\subseteqeq V$ and a $(1-\beta/2)p$-random subset $C'\subseteqeq C$. Fix $p'=(1-\beta/2)p$ and $\beta'=\beta/2$ and note that $p' \leq 1-\beta'$. By the above argument again, with high probability the conclusion of the ``$p'\leq (1-\beta')$'' version of the lemma applies to $V',C',p',\beta'$. Let $U$ be a set with $|U| \leq pn$. Choose $U'\subseteqeq U$ with $|U'|=|U|-\beta n/2$. Then $|U'| \leq pn - \beta n/2 \leq p'n$.
From the ``$p'\leq (1-\beta')$'' version of the lemma we get a $C'$-rainbow matching $M$ from $U'$ to $V'$ of size $|U'|-\beta n/2= |U|-\beta n$.
\epsilonilonnd{proof}
The following variant of Lemma~\ref{Lemma_nearly_perfect_matching} finds a rainbow matching which completely covers the deterministic set $U$. To achieve this we introduce a small amount of independence between the vertices/colours which are used in the matching.
\begin{lemma}[Perfect matchings]\label{Lemma_sat_matching_random_embedding}
Let $1\geq \gamma \gg n^{-1}$, let $p\in[0,1]$, and let $K_{2n+1}$ be $2$-factorized.
Suppose that we have disjoint sets $V_{dep}, V_{ind}\subseteqeq V(K_{2n+1})$, and $C_{dep}, C_{ind}\subseteqeq C(K_{2n+1})$ with $V_{dep}$ $p/2$-random, $C_{dep}$ $p$-random, and $V_{ind}, C_{ind}$ $\gamma$-random. Suppose that $V_{ind}$ and $C_{ind}$ are independent of each other. Then, the following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus (V_{dep}\cup V_{ind})$ of order $\leq p n$, there is a perfect $(C_{dep}\cup C_{ind})$-rainbow matching from $U$ into $(V_{dep}\cup V_{ind})$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\beta$ such that $1\geq \gamma \gg \beta\gg n^{-1}$.
With probability $1-o(n^{-1})$, we can assume the conclusion of
Lemma~\ref{Lemma_nearly_perfect_matching} holds for $V=V_{dep}, C=C_{dep}$ with $p=p, \beta=\beta, n=n$, and, by Lemma~\ref{Lemma_high_degree_into_random_set} applied to $V=V_{ind}, C=C_{ind}$ with $p=q=\gamma, n=n$ that the following holds. For each $v\in V(K_{2n+1})$, we have $|N_{C_{ind}}(v) \cap V_{ind}| \geq \gamma^2n>\beta n$. We will show that the property in the lemma holds.
Let then $U\subseteq V(K_{2n+1})\setminus (V_{dep}\cup V_{ind})$ have order $\leq p n$.
From the conclusion of Lemma~\ref{Lemma_nearly_perfect_matching}, there is a $C_{dep}$-rainbow matching $M_1$ of size $|U|-\beta n$ from $U$ to $V_{dep}$.
Since $|N_{C_{ind}}(v) \cap V_{ind}| >\beta n$ for each $v\in U\setminus V(M_1)$ we can construct a $C_{ind}$-rainbow matching $M_2$ into $V_{ind}$ covering $U\setminus V(M_1)$ (by greedily choosing this matching one edge at a time). The matching $M_1\cup M_2$ then satisfies the lemma.
\epsilonilonnd{proof}
We will also use a lemma about matchings using an exact set of colours.
\begin{lemma}[Matchings into random sets using specified colours]\label{Lemma_matching_into_random_set_using_specified_colours}
Let $p\gg q\gg\beta \gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $(p/2)$-random. With high probability, for any $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|\geq pn$, and any $C\subseteqeq C(K_{2n+1})$ with $|C|\leq q n$, there is a $C$-rainbow matching of size $|C|-\beta n$ from $U$ to $V$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\gamma$ such that $\beta\gg\gamma \gg n^{-1}$.
By Lemma~\ref{MPSvertex} (applied with $n'=2n+1$), with high probability, we have that, for any set $A\subseteq V(K_{2n+1})\setminus V$ with $|A|\geq pn\geq (2n+1)^{1/4}$, for all but at most $\gamma n$ colours there are at most $(1+\gamma)p|A|$ edges of that colour between $A$ and $V$. By Chernoff's bound, with high probability $|V|=(1\pm \gamma)pn$. We will show that the property in the lemma holds.
Fix then an arbitrary pair $U$, $C$ as in the lemma. Without loss of generality, $|U|=pn$. Let $M$ be a maximal $C$-rainbow matching from $U$ to $V$. We will show that $|M|\geq |C|-\beta n$, so that the matching required in the lemma must exist (by removing edges if necessary). Now, let $C'=C\setminus C(M)$. For each $c\in C'$, any edge between $U$ and $V$ with colour $c$ must have a vertex in $V(M)$, by maximality. Therefore, there are at most $2|V(M)|\leq 4qn$ edges of colour $c$ between $U$ and $V$. From the property from Lemma~\ref{MPScolour}, there are at most $\gamma n$ colours with more than $(1+\gamma)p|U|$ edges between $U$ and $V$. Therefore, using $|V|=(1\pm \gamma)pn$,
\[
|U||V|\leq 4qn|C'|+(2n+1)\gamma n+(1+\gamma)p|U|(n-|C'|)\leq (4qn-p^2n)|C'|+3\gamma n^2+ (1+\gamma)(1+2\gamma)|U||V|,
\]
and hence
\[
|C'|p^2n/2\leq |C'|(p^2-4q)n\leq ((1+\gamma)(1+2\gamma)-1)|U||V|+3\gamma n^2\leq 7\gamma n^2.
\]
It follows that $|C'|\leq 14\gamma n/p^2\leq \beta n$. Thus, $|M|= |C|-|C'|\geq |C|-\beta n$, as required.
\epsilonilonnd{proof}
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\subsection{Rainbow star forests}\label{Section_stars}
Here we develop techniques for embedding the high degree vertices of trees, based on our previous methods in~\cite{MPS}. We will do this by proving lemmas about large star forests in coloured graphs. In later sections, when we find rainbow trees, we isolate a star forest of edges going through high degree vertices, and embed them using the techniques from this section.
We start from the following lemma.
\begin{lemma}[\cite{MPS}]\label{Corollary_MPS_kdisjstars} Let $0<\epsilonilonpsilon<1/100$ and $\epsilonilonll\leq \epsilonilonpsilon^{2}n/2$. Let $G$ be an $n$-vertex graph with minimum degree at least $(1-\epsilonilonpsilon)n$ which contains an independent set on the distinct vertices $v_1,\ldots,v_\epsilonilonll$. Let $d_1,\ldots,d_\epsilonilonll\geq 1$ be integers satisfying $\sum_{i\in [\epsilonilonll]}d_i\leq (1-3\epsilonilonpsilon)n/k$,
and suppose $G$ has a locally $k$-bounded edge-colouring.
Then, $G$ contains disjoint stars $S_1,\ldots,S_\epsilonilonll$ so that, for each $i\in [\epsilonilonll]$, $S_i$ is a star centered at $v_i$ with $d_i$ leaves, and $\cup_{i\in [\epsilonilonll]}S_i$ is rainbow.
\epsilonilonnd{lemma}
The following version of the above lemma will be more convenient to apply.
\begin{lemma}[Star forest]\label{Lemma_star_forest}
Let $1\gg \epsilonilonta \gg\gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $F$ be a star forest with degrees $\geq 1$ whose set of centers is $I=\{i_1, \dots, i_{\epsilonilonll}\}$ with $e(F)\leq (1-\epsilonilonta)n$. Suppose we have disjoint sets $J, V\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ with $|V|\geq (1-\gamma)2n$, $|C|\geq (1-\gamma)n$ and $J=\{j_1, \dots, j_{\epsilonilonll}\}$.
Then, there is a $C$-rainbow copy of $F$ with $i_t$ copied to $j_t$, for each $t\in [\epsilonilonll]$, whose vertices outside of $I$ are copied to vertices in $V$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\epsilonilonpsilon$ such that $\epsilonilonta \gg \epsilonilonpsilon\gg\gamma$.
Let $G$ be the subgraph of $K_n$ consisting of edges touching $V$ with colours in $C$. Notice that $\delta(G)\geq (1-4\gamma)2n\geq (1-\epsilonilonpsilon)(2n+1)$. Since $V$ and $J$ are disjoint, $J$ contains no edges in $G$. Notice that, as $J$ and $V$ are disjoint, $\epsilonilonpsilon\gg \gamma$, and $|V|\geq (1-\gamma)2n$, we have $\epsilonilonll=|J|\leq 2\gamma n+1\leq \epsilonilonpsilon^2 (2n+1)/2$. Let $k=2$ and let $d_1, \dots, d_{\epsilonilonll}\geq 1$ be the degrees of $i_1,\ldots,i_\epsilonilonll$ in $F$. Notice that $\epsilonilonta \gg\epsilonilonpsilon$ implies $\sum_{i=1}^{\epsilonilonll}d_i=e(F)\leq (1-\epsilonilonta)n\leq (1-3\epsilonilonpsilon)(2n+1)/k$.
Applying Lemma~\ref{Corollary_MPS_kdisjstars} to $G$ with $\{v_1, \dots, v_{\epsilonilonll}\}=J$, $n'=2n+1$, $k=2$, $\epsilonilonll=\epsilonilonll$, $\epsilonilonpsilon=\epsilonilonpsilon$ we find the required rainbow star forest.
\epsilonilonnd{proof}
The above lemma can be used to find a rainbow copy of any star forest $F$ in a $2$-factorization as long as there are more than enough colours for a rainbow copy of $F$. However, we also want this rainbow copy to be suitably randomized. This is achieved by finding a star forest larger than $F$ and then randomly deleting each edge independently.
The following lemma is how we embed rainbow star forests in this paper. It shows that we can find a rainbow copy of any star forest so that the unused vertices and colours are $p$-random sets. Crucially, and unavoidably, the sets of unused vertices/colours depend on each other. This is where the need to consider dependent sets arises.
\begin{lemma}[Randomized star forest]\label{Lemma_randomized_star_forest}
Let $1\geq p, \alpha\gg \gamma\gg d^{-1}, n^{-1}$ and $\log^{-1} n\gg d^{-1}$. Let $K_{2n+1}$ be $2$-factorized.
Let $F$ be a star forest with degrees $\geq d$ with $e(F)= (1-p)n$ whose set of centers is $I=\{i_1, \dots, i_{\epsilonilonll}\}$. Suppose we have disjoint sets $V,J\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ with $|V|\geq (1-\gamma)2n$, $|C|\geq (1-\gamma)n$ and $J=\{j_1, \dots, j_{\epsilonilonll}\}$
Then, there is a randomized subgraph $F'$ which is with high probability a $C$-rainbow copy of $F$, with $i_t$ copied to $j_t$ for each $t$ and whose vertices outside $I$ are copied to vertices in $V$. Additionally there are randomized sets $U\subseteqeq V\setminus V(F'), D\subseteqeq C\setminus C(F')$ such that $U$ is a $(1-\alpha)p$-random subset of $V$ and $D$ is a $(1-\alpha)p$-random subset of $C$ (with $U$ and $D$ allowed to depend on each other).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\epsilonilonta$ such that $p, \alpha \gg \epsilonilonta\gg \gamma$.
Let $\hat F$ be a star forest obtained from $F$ by replacing every star $S$ with a star $\hat S$ of size $e(\hat S)=e(S)(1-\epsilonilonta)/(1-p)$. Notice that $e(\hat F)=(1-\epsilonilonta)n$, and, for each vertex $v$ at the centre of a star, $S$ say, in $F$, we have $d_{\hat F}(v)=e(\hat S)=e(S)(1-\epsilonilonta)/(1-p)=d_F(v)(1-\epsilonilonta)/(1-p)$. By Lemma~\ref{Lemma_star_forest}, there is a $C$-rainbow embedding $\hat F'$ of $\hat F$ with $i_t$ copied to $j_t$ for each $t$ and whose vertices outside $I$ are contained in $V$.
Let $U$ be a $(1-\alpha)p$-random subset of $V$. Let $F'=\hat F' \setminus U$. By Chernoff's Bound and $\log^{-1}n, p, \gamma \gg d^{-1}$, we have $\mathbb{P}(d_{F'}(v)< (1-\gamma)(1-p+\alpha p)d_{\hat F'}(v))\leq e^{-(1-p+\alpha p)\gamma^2d/3}\leq e^{-\log^5 n}$ for each center $v$ in $F$. Note that, for each centre $v$, $(1-\gamma)(1-p+\alpha p)d_{\hat F'}(v)=(1-\gamma)(1-p+\alpha p) d_F(v)(1-\epsilonilonta)/(1-p) \geq d_F(v)$, where this holds as $p,\alpha\gg \epsilonilonta\gg \gamma$. Taking a union bound over all the centers shows that, with high probability, $F'$ contains a copy of $F$.
Since $\hat F'$ was rainbow, we have that $C(\hat F')\setminus C(F')$ is a $(1-\alpha)p$-random subset of $C(\hat F')$.
Let $\hat{C}$ be a $(1-\alpha)p$-random subset of $C\setminus C(\hat F')$ and set $D=\hat{C}\cup (C(\hat F')\setminus C(F'))$. Now $D$ and $U$ are both $(1-\alpha)p$-random subsets of $C$ and $V$ respectively.
\epsilonilonnd{proof}
\subsection{Rainbow paths}\label{Section_paths}
Here we collect lemmas for finding rainbow paths and cycles in random subgraphs of $K_{2n+1}$. First we prove two lemmas about short paths between prescribed vertices. These lemmas are later used to incorporate larger paths into a tree.
\begin{lemma}[Short paths between two vertices]\label{Lemma_short_paths_between_two_vertices} Let $p\gg \mu \gg n^{-1}>0$ and suppose $K_{2n+1}$ is $2$-factorized. Let $V\subseteq V(K_{2n+1})$ and $C\subseteq C(K_{2n+1})$ be $p$-random and independent. Then, with high probability, for each pair of distinct vertices $u,v\in V(K_{2n+1})$ there are at least
$\mu n$ internally vertex-disjoint $u,v$-paths with length $3$ and internal vertices in $V$ whose union is $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof}
Choose $p\gg \mu \gg n^{-1}>0$.
Randomly partition $C=C_1\cup C_2\cup C_{3}$ into three $p/3$-random sets and $V=V_1\cup V_2$ into two $p/2$-random sets. With high probability the following simultaneously hold.
$\bullet$ By Lemma~\ref{Lemma_MPS_boundrandcolour} applied to $C_3$ with $p=p/3, \epsilonilonpsilon=1/2, n=2n+1$, for any disjoint vertex sets $U,V$ of order at least $p^2n/10\geq (2n+1)^{3/4}$, we have $e_{C_3}(U,V)\geq p|U||V|/6\geq 10^{-3}p^5n^2$.
$\bullet$ By Lemma~\ref{Lemma_high_degree_into_random_set} applied to $C_i, V_j$ with $p=p/2, q=p/3, n=n$ we have $|N_{C_{i}}(v)\cap V_j| \geq p^2n/6$ for every $v\in V(K_{2n+1})$, $i\in [3]$, and $j\in [2]$.
We claim the property holds. Indeed, pick an arbitrary distinct pair of vertices $u,v\in V(K_{2n+1})$. Let $M$ be a maximum $C_3$-rainbow matching between $N_{C_{1}}(v)\cap (V_1\setminus \{u\})$ and $N_{C_{2}}(u)\cap (V_2\setminus \{v\})$ (these sets have size at least $p^2n/24$). Each of the $2e(M)$ vertices in $M$ has $2n$ neighbours in $K_{2n+1}$, and each colour in $M$ is on $2n+1$ edges in $K_{2n+1}$. The number of edges of $K_{2n+1}$ sharing a vertex or colour with $M$ is thus $\leq 7ne(M)$. By maximality, and the property from Lemma~\ref{Lemma_MPS_boundrandcolour}, we have $7ne(M)\geq e_{C_3}(U,V) \geq 10^{-3}p^5n^2$, which implies that $e(M)\geq 10^{-4}p^5n \geq 4\mu n$.
For any edge $v_1v_2$ in the matching $M$, the path $uv_1v_2v$ is a rainbow path. In the union of these paths, the only colour repetitions can happen at $u$ or $v$. Since $K_{2n+1}$ is $2$-factorized, there is a subfamily of $\mu n$ paths which are collectively rainbow.
\epsilonilonnd{proof}
We can use this lemma to find many disjoint length $3$ connecting paths.
\begin{lemma}[Short connecting paths]\label{Lemma_few_connecting_paths} Let $p\gg q \gg n^{-1}>0$ and let $K_{2n+1}$ be $2$-factorized. Let $V$ be a $p$-random set of vertices, and $C$ a $p$-random set of colours independent from $V$.
Then, with high probability, for any set of $\{x_1, y_1, \dots, x_{q n}, y_{q n}\}$ of vertices,
there is a collection $P_1, \dots, P_{q n}$ of vertex-disjoint paths with length $3$, having internal vertices in $V$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [qn]$, and $P_1\cup \dots\cup P_{q n}$ is $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof}
By Lemma~\ref{Lemma_short_paths_between_two_vertices} applied to $C,V$ with $p=p, \mu=10q, n=n$, with high probability, between any $x_i$ and $y_i$, there is a collection of $10q n$ internally vertex disjoint $x_i,y_i$-paths, which are collectively $C$-rainbow, and internally contained in $V$. By choosing such paths greedily one by one, making sure never to repeat a colour or vertex, we can find the required collection of paths.
\epsilonilonnd{proof}
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\section{The finishing lemma in Case A}\label{sec:finishA}
The proof of our finishing lemmas uses \epsilonilonmph{distributive} absorption, a technique introduced by the first author in \cite{montgomery2018spanning}. For the finishing lemma in Case A, we start by constructing colour switchers for sets of $\leq 100$ colours of $C$. These are $|C|$ perfect rainbow matchings, each from the same small set $X$ into a larger set $V$ which use the same colour except for one different colour from $C$ per matching (see Lemma~\ref{absorbA}). This gives us a small amount of local variability, but we can build this into a global variability property (see Lemma~\ref{absorbAmacro}). This will allow us to choose colours to use from a large set of colours, but not all the colours, so we will need to find matchings which ensure any colours outside of this are used (see Lemma~\ref{colourcover}). To find the switchers we use a small proportion of colours in a random set. We will have to cover the colours not used in this, with no random properties for the colours remaining (see Lemma~\ref{Lemma_saturating_matching_lemma}). We put this all together to prove Lemma~\ref{lem:finishA} in Section~\ref{sec:finishAfinal}.
\subsection{Colour switching with matchings}\label{sec:switcherspaths}
We start by constructing colour switchers using matchings.
\begin{lemma}\label{absorbA} Let $1/n\ll \beta \ll\xi, \mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$-random subsets which are independent. With high probability, the following holds.
Let $C,\bar{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},Z\subseteq V(K_{2n+1})$ satisfy $|\bar{{C}}|,|\bar{V}|\leq \beta n$, $|C|\leq 100$ and suppose that $(X,Z)$ is $(\xi n)$-replete.
Then, there are sets $X'\subseteq X\setminus \bar{V}$, $C'\subseteq D_0\setminus \bar{C}$ and $V'\subseteq (V_0\cup Z)\setminus \bar{V}$ with sizes $|C|$, $|C|-1$ and $\leq 3|C|$ respectively such that, for every $c\in C$, there is a perfect $(C'\cup \{c\})$-matching from $X'$ to $V'$.
\epsilonilonnd{lemma}
\begin{proof}
With high probability, by Lemma~\ref{lem:goodpairs} we have the following property.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}}]
\item For each distinct $u,v\in V(K_{2n+1})$, there are at least $100\beta n$ colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$.\label{prop:goodpairs}
\epsilonilonnd{enumerate}
We will show the property in the lemma holds. Let $C,\bar{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},Z\subseteq V(K_{2n+1})$ be sets with $|\bar{{C}}|,|\bar{V}|\leq \beta n$, $|C|\leq 100$ and suppose that $(X,Z)$ is $(\xi n)$-replete.
Let $\epsilonilonll=|C|\leq 100$ and label $C=\{c_1,\ldots,c_\epsilonilonll\}$.
For each $i\in [\epsilonilonll]$, using that there are at least $\xi n$ edges with colour $C$ between $X$ and $Z$, and $\xi\gg \beta$, $\epsilonilonll\leq 100$ and $|\bar{V}|\leq \beta n$, pick a vertex $x_i\in X\setminus (\bar{V}\cup \{x_1,\ldots,x_{i-1}\})$ which has a $c_i$-neighbour $y_i\in Z\setminus (\bar{V}\cup \{y_1,\ldots,y_{i-1}\})$.
For each $1\leq i\leq \epsilonilonll-1$, using~\ref{prop:goodpairs}, pick a colour $d_i\in D_0\setminus (\bar{C}\cup C\cup \{d_1,\ldots,d_{i-1}\})$
and (not necessarily distinct) vertices $z_i,z'_i\in V_0\setminus (\bar{V}\cup \{y_1,\ldots,y_{\epsilonilonll},z_1,\ldots,z_{i-1},z_1',\ldots,z_{i-1}'\})$ such that $x_iz_i$ and $x_{i+1}z'_{i}$ are both colour $d_i$. Let $C'=\{d_1,\ldots,d_{\epsilonilonll-1}\}$, $X'=\{x_1,\ldots,x_{\epsilonilonll}\}$ and $V'= \{y_1,\ldots,y_{\epsilonilonll},z_1,\ldots,z_{\epsilonilonll-1},z_1',\ldots,z_{\epsilonilonll-1}'\}$. See Figure~\ref{Figure_Matching_Switching} for an example of the edges that we find.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{matchingswitching.pdf}
\caption{An example of the graph we find in Lemma~\ref{absorbA} when $\epsilonilonll=5$. The key property it has is that for any colour $c_i$, there is a matching covering $x_1, \dots, x_{\epsilonilonll}$ using $c_i$ and all the colours $d_{1}, \dots, d_{\epsilonilonll-1}$.}
\label{Figure_Matching_Switching}
\epsilonilonnd{figure}
Now, for each $j\in [\epsilonilonll]$, let $M_j=\{x_jy_j\}\cup \{x_iz_i:i<j\}\cup \{x_{i+1}z_i':j\leq i\leq \epsilonilonll-1\}$. Note that $M_j$ is a perfect $(C\cup \{c_j\})$-matching from $X'$ to $V'$, and thus $X'$, $C'$ and $V'$ have the property required.
\epsilonilonnd{proof}
\subsection{Distributive absorption with matchings}
We now put our colour switchers together to create a global flexibility property. To do this, we find certain disjoint colour switchers, governed by a suitable \epsilonilonmph{robustly matchable bipartite graph}.
This is a bipartite graph which has a lot of flexibility in how one of its parts can be covered by matchings. It exists by the following lemma.
\begin{lemma}[Robustly matchable bipartite graphs, \cite{montgomery2018spanning}]\label{Lemma_H_graph}
There is a constant $h_0 \in \mathbb N$ such that, for every $h \geq h_0$, there exists
a bipartite graph $H$ with maximum degree at most $100$ and vertex classes $X$ and $Y \cup Y'$, with
$|X| = 3h$, and $|Y| = |Y'| = 2h$, so that the following is true. If $Y_0 \subseteqeq Y'$ and $|Y_0 | = h$, then
there is a matching between $X$ and $Y \cup Y_0$.
\epsilonilonnd{lemma}
\begin{lemma}\label{absorbAmacro} Let $1/n\ll \beta \ll \epsilonilonta\ll \xi,\mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$- and $\epsilonilonta$-random respectively, and suppose that they are independent. With high probability, the following holds.
Suppose $X,Z\subseteq V(K_{2n+1})$ are disjoint subsets such that $(X,Z)$ is $(\xi n)$-replete, $\alpha\leq \beta$ and $C\subseteq C(K_{2n+1})\setminus D_0$ is a set of at most $2\beta n$ colours. Then, there is a set $X_0$ of $|D_0|+\alpha n$ vertices in $X$ such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect $(D_0\cup C')$-rainbow matching from $X_0$ into $V_0\cup Z$.
\epsilonilonnd{lemma}
\begin{proof}
With high probability, by Lemmas~\ref{Lemma_Chernoff} and~\ref{absorbA} (applied with $\beta'=10^3\beta, \mu'=\epsilonilonta$, $\xi'=\xi/2$, and an $\epsilonilonta$-random subset of $V(K_{2n+1})$ contained in $V_0$) we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $\epsilonilonta n/2\leq |D_0|\leq 2\epsilonilonta n$.\label{aa2}
\item Let $\bar{C},\hat{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},\bar{Z}\subseteq V(K_{2n+1})$ satisfy $|\bar{{C}}|,|\bar{V}|\leq 10^3\beta n$, $|\hat{C}|\leq 100$ and suppose that $(X,\bar{Z})$ is $(\xi n/2)$-replete.
Then, there are sets $X'\subseteq Y\setminus \bar{V}$, $C'\subseteq D_0\setminus \bar{C}$ and $V'\subseteq V\cup \bar Z \setminus \bar{V}$ with sizes $|\hat{C}|$, $|\hat{C}|-1$ and $\leq 3|\hat{C}|$ such that, for every $c\in \hat{C}$, there is a perfect $(C'\cup \{c\})$-matching from $Y'$ to $V'$.\label{aa3}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. For this, let $X,Z\subseteq V(K_{2n+1})$ be disjoint subsets such that $(X,Z)$ is $(\xi n)$-replete, let $\alpha\leq \beta$ and let $C\subseteq C(K_{2n+1})\setminus D_0$ be a set of at most $2\beta n$ colours.
Let $h=2\beta n$, and, using \ref{aa2}, pick a set $D_1\subseteq D_0$ of $(3h-\alpha n)$ colours.
Noting that $|D_1|\geq 2h$ and $|D_1\cup C|\leq 4h$, we can define sets $Y,Y'$ of order $2h$ with $Y\subseteqeq D_1$ and $D_1\cup C\subseteqeq Y\cup Y'$ (where any extra elements of $Y'$ are arbitrary dummy colours which will not be used in the arguments).
Using Lemma~\ref{Lemma_H_graph}, let $H$ be a bipartite graph with vertex classes $[3h]$ and $Y\cup Y'$, which has maximum degree at most 100, and is such that for any $\bar{Y}\subseteqeq Y'$ with $|\bar{Y}|=h$, there is a perfect matching between $[3h]$ and $Y\cup \bar{Y}$. In particular, since $|D_1|=3h-\alpha n$ and $Y\subseteqeq D_1$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\addtocounter{enumi}{2}
\item For each set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect matching between $[3h]$ and $D_1\cup C'$ in $H$. \label{Hmatch}
\epsilonilonnd{enumerate}
For each $i\in [3h]$, let $D_i=N_H(i)\cap C(K_{2n+1})$ (i.e.\ $D_i$ is the set of non-dummy colours in $N_H(i)$). Using \ref{aa3} repeatedly, find sets $X_i\subseteq X\setminus (X_1\cup\ldots\cup X_{i-1})$, $C_i\subseteq D_0\setminus (D_1\cup C_1\cup\ldots\cup C_{i-1})$ and $V_i\subseteq (V_0\cup Z)\setminus (V_1\cup \ldots\cup V_{i-1})$,
with sizes $|D_i|$, $|D_i|-1$ and at most $300$, such that the following holds.
(To see that we can repeat this application of \ref{aa3} this many times, note that $h =2\beta n$ and $\xi \gg \beta$ and so at each application we delete $O(h)$ vertices from $X$ or $Z$, leaving a pair which is still $(\xi n/2)$-replete.)
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\addtocounter{enumi}{3}
\item For every $c\in D_i$, there is a perfect $(C_i\cup\{c\})$-rainbow matching from $X_i$ into $V_i$.
\label{propabsorb}
\epsilonilonnd{enumerate}
Greedily, using \ref{aa2}, that $h=2\beta n$, $\beta,\epsilonilonta\ll \xi$ and $(X,Z)$ is $(\xi n)$-replete, find a $(D_0\setminus (D_1\cup C_1\cup\ldots\cup C_{3h}))$-rainbow matching $M$ with $|D_0\setminus (D_1\cup C_1\cup \ldots\cup C_{3h})|$ edges from $X\setminus (X_1\cup\ldots\cup X_{3h})$ into $Z\setminus (X_1\cup\ldots\cup X_h)$.
Let $X_0=(X\cap V(M))\cup X_1\cup \ldots\cup X_{3h}$. Note that $|X_0|=|D_0|-|D_1|-\sum_{i=1}^{3h}(|X_i|-|C_i|)=|D_0|-|D_1|+3h=|D_0|+\alpha n$.
We claim this has the property required. Indeed, suppose $C'\subseteq C$ is a set of $\alpha n$
colours. Using~\ref{Hmatch}, let $M'$ be a perfect matching between $[3h]$ and $D_1\cup C'$ in $H$, and label $D_1\cup C'=\{c_1,\ldots,c_{3h}\}$ so that, for each $i\in [3h]$, $c_i$ is matched to $i$ in $M'$. Note that the colours $c_i$ are not dummy colours.
By the definition of each $D_i$, $c_i\in D_i$ for each $i\in [3h]$.
By \ref{propabsorb}, for each $i\in [3h]$, there is a perfect $(C_i\cup \{c_i\})$-matching, $M_i$ say, from $X_i$ to $V_i$. Then, $M\cup M_1\cup \ldots\cup M_{3h}$ is a perfect matching from $X_0$ into $V_0\cup Z$ which is $(C(M)\cup D_1\cup C'\cup C_1\cup\ldots\cup C_{3h})$-rainbow, and hence $(D_0\cup C')$-rainbow, as required.
\epsilonilonnd{proof}
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0.54.15
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\subsection{Covering small colour sets with matchings}
We find perfect rainbow matchings covering a small set of colours using the following lemma.
\begin{lemma}\label{colourcover} Let $1/n\ll \nu\ll \lambda\ll 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $X,V\subseteq V(K_{2n+1})$ are disjoint, $(X,V)$ is $(10\nu n)$-replete and $|X|\leq \lambda n$. Suppose that $C\subseteq C(K_{2n+1})$ is such that every vertex $v\in V(K_{2n+1})$ has at least $3\lambda n$ colour-$C$ neighbours in $V$.
Then, given any set $C'\subseteq C(K_{2n+1})$ with at most $\nu n$ colours, there is a perfect $(C'\cup C)$-rainbow matching from $X$ to $V$ which uses every colour in $C'$.
\epsilonilonnd{lemma}
\begin{proof} Using that $(X,V)$ is $(10\nu n)$-replete, greedily find a matching $M_1$ with $|C'|$ edges from $X$ to $V$ which is $C'$-rainbow. Then, using that every vertex $v\in V(K_{2n+1})$ has at least $3\lambda n$ colour-$C$ neighbours in $V$, greedily find a perfect $C$-rainbow matching $M_2$ from $X\setminus V(M_1)$ to $V\setminus V(M_1)$. This is possible as, when building $M_2$ greedily, each vertex in $X\setminus V(M_1)$ has at most $2|X|\leq 2\lambda n$ neighbouring edges with colour used in $C\cap C(M_1\cup M_2)$ (as the colouring is $2$-bounded) and at most $|X|\leq \lambda n$ colour-$C$ neighbours in $V(M_1)\cup V(M_2)$.
The matching $M_1\cup M_2$ then has the required property.
\epsilonilonnd{proof}
\subsection{Almost-covering colours with matchings}
We find rainbow matchings using almost all of a set of colours using the following lemma.
\begin{lemma}\label{Lemma_saturating_matching_lemma}
Let $1\geq p\gg q\gg \gamma,\epsilonilonta \gg \nu\gg 1/n$ and let $K_{2n+1}$ be $2$-factorized.
Suppose we have disjoint sets $V, V_0\subseteqeq V(K_{2n+1})$, and $D,D_0\subseteqeq C(K_{2n+1})$ with $V$ $p$-random, $D$ $q$-random, and $V_0, D_0$ $\gamma$-random. Furthermore, suppose that $V_0$ and $D_0$ are independent of each other. Then, the following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus (V\cup V_0)$ with at most $(q+\gamma+\epsilonilonta-\nu)n$ vertices, and any $C\subseteq C(K_{2n+1})$ with $D\cup D_0\subseteq C$ with $|C|\geq |U|+\nu n$, there is a perfect $C$-rainbow matching from $U$ into $V\cup V_0$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\beta$ such that $\nu\gg \beta\gg n^{-1}$.
Partition $V=V_1\cup V_2$ so that $V_1$ and $V_2$ are $(p/2)$-random. Partition $D_0=D_1\cup D_2$ with $D_1$ $(\gamma-\nu/4)$-random and $D_2$ $(\nu/4)$-random. We now find properties \ref{grape1} -- \ref{grape4}, which all hold with probability $1-o(n^{-1})$, as follows.
By Lemma~\ref{Lemma_matching_into_random_set_using_specified_colours} with $V=V_1$, $p'=q/2$, $q=\gamma+2\epsilonilonta$, $\beta=\beta$, and $n=n$, we have the following.
\addtocounter{propcounter}{1}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any $U\subseteq V(K_{2n+1})\setminus V_1$ with $|U|\geq qn/2$ and any $C'\subseteq C(K_{2n+1})$ with $|C'|\leq (\gamma +2\epsilonilonta)n$, there is a $C'$-rainbow matching of size $|C'|-\beta n$ from $U$ into $V_1$. \label{grape1}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_nearly_perfect_matching} with $V=V_2$, $C=D$, $p=q$, $\beta=\beta$, and $n=n$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\stepcounter{enumi}
\item For any $U\subseteq V(K_{2k+1})\setminus V_2$ with $|U|\leq q n$, there is a $D$-rainbow matching from $U$ into $V_2$ with size $|U|-\beta n$.\label{grape2b}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_high_degree_into_random_set} with $V=V_0$, $C=D_2$, $p=\gamma$, $q=\nu/4$, and $n=n$ we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item Every vertex $v\in V(K_{2n+1})$ has $|N_{D_2}(v)\cap V_0|\geq \gamma\nu n/4$.\label{grape3}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_Chernoff}, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $|D_1|=(\gamma-\nu/4\pm \beta)n\leq \gamma n$, $|D|=(q\pm \beta)n$ and $|D_2| \leq \nu n/3$.\label{grape4}
\epsilonilonnd{enumerate}
We now show that the property in the lemma holds. For this, let $U\subseteqeq V(K_{2n+1})\setminus (V\cup V_0)$ have at most $(q+\gamma+\epsilonilonta-\nu)n$ vertices, and let $C\subseteq C(K_{2n+1})$ satisfy $D\cup D_0\subseteq C$ and $|C|\geq |U|+\nu n$.
Note that we can assume that $|C|=|U|+\nu n$, so that, using \ref{grape4}, $|C\setminus (D \cup D_2)|\leq |U|+\nu n -|D| \leq (q+\gamma+\epsilonilonta-\nu)n+\nu n-(q-\beta)n= (\gamma+\epsilonilonta +\beta)n\leq (\gamma+2\epsilonilonta)n$. Notice that, by \ref{grape4} and as $D\subseteq C$, we have $|U|\geq |C|-\nu n\geq q n/2$.
Therefore, by \ref{grape1}, there is a $(C\setminus (D \cup D_2))$-rainbow matching $M_1$ of size $|C\setminus (D \cup D_2)|-\beta n$ from $U$ into $V_1$. Now, using \ref{grape4}, $|U\setminus V(M_1)|=|U|-|C\setminus (D \cup D_2)|+\beta n\leq |D|+|D_2|-\nu n+\beta n\leq (q+\beta)n+\nu n/3-\nu n+\beta n\leq q n$. By \ref{grape2b}, there is a $D$-rainbow matching $M_2$ of size $|U\setminus V(M_1)| -\beta n$ from $U\setminus V(M_1)$ into $V_2$.
Notice that $|U\setminus (V(M_1)\cup V(M_2))|\leq \beta n$.
From \ref{grape3}, we have that $|N_{D_2}(v)\cap V_0|\geq \gamma \nu n/4> 3 \beta n \geq 3|U\setminus (V(M_1)\cup V(M_2))|$ for every vertex $v\in V(K_{2n+1})$. By greedily choosing neighbours of $u\in U\setminus (V(M_1)\cup V(M_2))$ one at a time (making sure to never repeat colours or vertices) we can find a $D_2$-rainbow matching $M_3$ into $V_0$ covering $U\setminus (V(M_1)\cup V(M_2))$. Indeed, when we seek a new colour-$D_2$ neighbour of a vertex $u\in U\setminus (V(M_1)\cup V(M_2))$ there are at most 2 neighbours of $u$ of each colour used from $D_2$ in $V_0$, ruling out at most $2|U\setminus (V(M_1)\cup V(M_2))|$ colour-$D_2$ neighbours of $u$ in $V_0$, while at most $|U\setminus (V(M_1)\cup V(M_2))|$ vertices have been used in $V_0$ in the matching.
Now the matching $M_1\cup M_2\cup M_3$ satisfies the lemma.
\epsilonilonnd{proof}
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0.54.16
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\subsection{Proof of the finishing lemma in Case A}\label{sec:finishAfinal}
Finally, we can put all this together to prove Lemma~\ref{lem:finishA}.
\begin{proof}[Proof of Lemma~\ref{lem:finishA}]
Pick $\nu,\lambda,\beta$ and $\alpha$ so that $1/n\ll \nu \ll\lambda\ll\beta\ll \alpha\ll \xi$ and recall that $\xi \ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll p\leq 1$. Let $V_1',V_2',V_3'$ be disjoint $(p/3)$-random subsets in $V$ and let $W_1,W_2,W_3$ be disjoint $(\mu/3)$-random subsets in $V_0$.
Let $D_1$, $D_2$, $D_3$ be disjoint $(\mu/3)$-, $\beta$- and $\alpha$-random disjoint subsets in $C_0$ respectively.
We now find properties \ref{mouse11}--\ref{mouse44}, which collectively hold with high probability as follows.
First note that $(1-\epsilonilonta)\epsilonilonps+(\mu/3)+\epsilonilonta-\nu \geq \epsilonilonps$. Thus, by Lemma~\ref{Lemma_saturating_matching_lemma} applied with $V=V_1'$, $V_0=W_1$, $D=C$, $D_0=D_1$, $p'=p/3$, $q=(1-\epsilonilonta)\epsilonilonpsilon$, $\gamma=\mu/3$, $\epsilonilonta=\epsilonilonta$ and $\nu=\nu$ we get the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any $U\subseteq V(K_{2n+1})\setminus (V'_1\cup W_1)$ with $|U|\leq \epsilonilonps n$ and any set of colours $C'\subseteq C(K_{2n+1})$ with $C\cup D_1\subseteq C'$, and $|C'| \geq |U|+\nu n$, there is a perfect $C'$-rainbow matching from $U$ into $V_1'\cup W_1$.\label{mouse11}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_high_degree_into_random_set}, we get the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item Each $v\in V(K_{2n+1})$ has at least $\beta^2 n$ colour-$D_2$ neighbours in $W_2$.\label{mouse2}
\epsilonilonnd{enumerate}
By Lemma~\ref{absorbAmacro} applied with $V_0=W_3$,
$D_0=D_3$, $\mu'=\mu/3$, $\xi'=\xi/3$, $\epsilonilonta'=\alpha, \alpha'=\gamma$ and $\beta'=2\beta$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any disjoint vertex sets $Y,Z\subseteq V(K_{2n+1})\setminus W_3$ for which $(Y,Z)$ is $(\xi n/3)$-replete, any $\gamma\leq 2\beta$ and any set $\hat{C}\subseteq C(K_{2n+1})\setminus D_3$ of at most $4\beta n$ colours, the following holds. There is a subset $Y'\subseteq Y$ of $|D_3|+\gamma n$ vertices so that, for any $C'\subseteq \hat{C}$ with $|C'|=\gamma n$, there is a perfect $(D_3\cup C')$-rainbow matching from $Y'$ into $Z\cup W_3$.\label{mouse3}
\epsilonilonnd{enumerate}
Finally, by Lemma~\ref{Lemma_Chernoff}, the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $2\beta n\geq |D_2|\geq \beta n/2$ and $2\alpha n\geq |D_3|\geq \alpha n/2$.\label{mouse44}
\epsilonilonnd{enumerate}
We will now show that the property in the lemma holds.
Let then $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ be disjoint and let $D\subseteq C(K_{2n+1})$, so that $|X|=|D|=\epsilonilonps n$, $C_0\cup C\subseteq D$ and $(X,Z)$ is $(\xi n)$-replete. Let $Z_1$, $Z_2$ be disjoint $(1/2)$-random subsets of $Z$. Note that, by Lemma~\ref{Lemma_Chernoff}, $(X,Z_1)$ and $(X,Z_2)$ are with high probability $(\xi/3)$-replete. Thus, we can pick an instance of $Z_1$ and $Z_2$ for which this holds. Let $V_1=V_1'\cup W_1$, $V_2=V_2'\cup W_2\cup Z_1$ and $V_3=V_3'\cup W_3\cup Z_2$.
Set $\gamma=(|D_2|/n)-\lambda$, so that, by \ref{mouse44}, $2\beta\geq \gamma>0$. Then, by \ref{mouse44} and \ref{mouse3} applied with $\hat{C}=D_2$, $Y=X_1, Z=Z_2$ there is a set $X_3\subseteq X$ with size $|D_3|+\gamma n=|D_3|+|D_2|-\lambda n$ and the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{4}
\item For any $C'\subseteq D_2$ with $|C'|=\gamma n=|D_2|-\lambda n$, there is a perfect $(D_3\cup C')$-rainbow matching from $X_3$ into $V_3$.\label{mouse4}
\epsilonilonnd{enumerate}
Let $C_1=D\setminus (D_2\cup D_3)$, $C_2=D_2$ and $C_3=D_3$, and note that $C\cup D_1\subseteq C_1$.
Let $X'=X\setminus X_3$, so that $|X'|=(\epsilonilonps +\lambda)n-|D_3|-|D_2|=|C_1|+\lambda n$. Using \ref{mouse44}, we have $|X_3|\leq |D_2|+|D_3|\leq \xi n/10$, and hence $(X',Z_1)$ is $(\xi n/6)$-replete.
Let $X_2'\subseteq X_2$ be a $(\lambda n/|X'|)$-random subset of $X'$. By Lemma~\ref{Lemma_Chernoff}, $|X_2'|\leq (\lambda+\nu)n$ and $(X_2',Z_1)$ is $(10\nu n)$-replete. Choose disjoint sets $X_1$ and $X_2$ of $X'$ with size $|X'|-(\lambda+\nu)n=|C_1|-\nu n$ and $(\lambda+\nu)n$ respectively, and so that $X_2'\subseteq X_2$. Note that $(X_2,Z_1)$ is $(10\nu n)$-replete. Therefore, by Lemma~\ref{colourcover} (with $\nu=\nu$, $\lambda'=\lambda+\nu \ll \beta$, $X=X_2, V=V_2$, and $C=C_2=D_2$) and \ref{mouse2}, the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{5}
\item For any set $C'\subseteq C(K_n)$ of at most $\nu n$ colours, there is a perfect $(C'\cup C_2)$-rainbow matching from $X_2$ to $V_2$ which uses every colour in $C'$.\label{mouse5}
\epsilonilonnd{enumerate}
We thus have partitions $X=X_1\cup X_2\cup X_3$, $C=C_1\cup C_2\cup C_3$ and $V=V_1\cup V_2\cup V_3$, for which, by \ref{mouse11}, \ref{mouse5} and \ref{mouse4}, we have that
\ref{propp1}--\ref{propp3} hold. Therefore, by the discussion after \ref{propp1}--\ref{propp3}, we have a perfect $C$-rainbow matching from $X$ into $V$.
\epsilonilonnd{proof}
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0.54.17
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\section{The finishing lemma in Case B}\label{sec:finishB}
Our proof of the finishing lemma in Case B has the same structure as the finishing lemma in Case A, except we first construct the colour switchers in the ND-colouring in two steps, before showing this can be done with random vertices and colours. In overview, in this section we do the following, noting the lemmas in which the relevant result is given and the comparable lemmas in Section~\ref{sec:finishA}.
\begin{itemize}
\item Lemma~\ref{lem-switchpath}: We construct colour switchers in the ND-colouring for any pair of colours $(c_1,c_2)$ -- that is, two short rainbow paths with the same length between the same pair of vertices, whose colours are the same except that one uses $c_1$ and one uses $c_2$.
\item Lemma~\ref{lem-absorbpath}: We use Lemma~\ref{lem-switchpath} to construct a similar colour switcher in the ND-colouring that can use 1 of a set of 100 colours.
\item Lemma~\ref{lem-randabsorbpath}: We show that, given a pair of independent random vertex and colour sets, many of these switchers use only vertices and (non-switching) colours in the subsets (cf.\ Lemma~\ref{absorbA}).
\item Lemma~\ref{absorbBmacro}: We use distributive absorption to convert this into a larger scale absorption property (cf.\ Lemma~\ref{absorbAmacro}).
\item Lemma~\ref{colourcoverB}: We embed paths while ensuring that an (arbitrary) small set of colours is used (cf.\ Lemma~\ref{colourcover}).
\item Lemma~\ref{finishingB}: We embed paths using almost all of a set of colours, in such a way that this can reduce the number of `non-random' colours (cf.\ Lemma~\ref{Lemma_saturating_matching_lemma}).
\item Finally, we put this all together to prove Lemma~\ref{lem:finishB}, the finishing lemma in Case B (cf.\ the proof of Lemma~\ref{lem:finishA}).
\epsilonilonnd{itemize}
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\subsection{Colour switching with paths}\label{sec:switchers}
We start by constructing colour switchers in the ND-colouring capable of switching between 2 colours.
We use switchers consisting of two paths with length 7 between the same two vertices. By using one path or the other we can choose which of two colours $c_1$ and $c_2$ are used, as the other colours used appear on both paths.
\begin{defn}
In a complete graph $K_{2n+1}$ with vertex set $[2n+1]$, let $(x_0,a_1,\ldots,a_\epsilonilonll)$ denote the path with length $\epsilonilonll$ with vertices $x_0x_1\ldots x_\epsilonilonll$ where, for each $i\in [\epsilonilonll]$, $x_i=x_{i-1}+a_i\pmod 2n+1$.
\epsilonilonnd{defn}
\begin{figure}[h]
\begin{center}
{
\begin{tikzpicture}[scale=0.7]
\draw [red,thick] (-1,0) -- (0,0);
\draw (-0.5,-0.3) node {$i$};
\draw [blue,thick] (4,1) -- (0,0);
\draw (2,0.75) node {$c_1$};
\draw [green,thick] (4,1) -- (6,1);
\draw (5,1.25) node {$d_1$};
\draw [magenta,thick] (13,1) -- (6,1);
\draw (9.5,1.25) node {$d_2$};
\draw [purple,thick] (13,1) -- (10,0.5);
\draw (11.5,0.45) node {$d_3$};
\draw [orange,thick] (5,0) -- (10,0.5);
\draw (7.5,0.55) node {$d_4$};
\draw [brown,thick] (5,0) -- (18,0);
\draw (15,-0.3) node {$y-i-c_1-k-x$};
\draw [green,thick] (5,0) -- (7,-0.5);
\draw (6,-0.55) node {$d_1$};
\draw [magenta,thick] (14,-1) -- (7,-0.5);
\draw (10.5,-0.45) node {$d_2$};
\draw [purple,thick] (14,-1) -- (11,-1);
\draw (12.5,-1.25) node {$d_3$};
\draw [orange,thick] (6,-1) -- (11,-1);
\draw (8.5,-1.25) node {$d_4$};
\draw [cyan,thick] (6,-1) -- (0,0);
\draw (3,-0.75) node {$c_2$};
\draw [fill=black] (-1,0) circle [radius = 0.1cm];
\draw [fill=black] (18,0) circle [radius = 0.1cm];
\draw [fill=black] (0,0) circle [radius = 0.1cm];
\draw [fill=black] (4,1) circle [radius = 0.1cm];
\draw [fill=black] (6,1) circle [radius = 0.1cm];
\draw [fill=black] (13,1) circle [radius = 0.1cm];
\draw [fill=black] (10,0.5) circle [radius = 0.1cm];
\draw [fill=black] (5,0) circle [radius = 0.1cm];
\draw [fill=black] (7,-0.5) circle [radius = 0.1cm];
\draw [fill=black] (6,-1) circle [radius = 0.1cm];
\draw [fill=black] (11,-1) circle [radius = 0.1cm];
\draw [fill=black] (14,-1) circle [radius = 0.1cm];
\draw (18.5,0) node {$y$};
\draw (-1.5,0) node {$x$};
\epsilonilonnd{tikzpicture}
}
\epsilonilonnd{center}
\caption{Two $x,y$-paths used to switch colours between $c_1$ and $c_2$.}\label{colourswitcher}
\epsilonilonnd{figure}
\begin{lemma}[1 in 2 colour switchers]\label{lem-switchpath}
Let $1\gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. Suppose we have a pair of distinct vertices $x,y\in V(K_{2n+1})$, a pair of distinct colours $c_1,c_2\in C(K_{2n+1})$, and sets $X\subseteq V(K_{2n+1})$ and $C\subseteq C(K_{2n+1})$ with $|X|\leq n/25$ and $|C|\leq n/25$.
Then, there is some set $C'\subseteq C(K_{2n+1})\setminus (C\cup\{c_1,c_2\})$ with size $6$ so that, for each $i\in \{1,2\}$, there is a $(C'\cup\{c_i\})$-rainbow $x,y$-path with length 7 and interior vertices in $V(K_{2n+1})\setminus X$.
\epsilonilonnd{lemma}
\begin{proof}
Since $2n+1$ is odd we can relabel $c_1$ and $c_2$ such that $c_1+2k=c_2$ for some $k\in [n]$ (here and later in this proof all the additions are$\pmod{2n+1}$). We will construct a switcher as depicted in Figure~\ref{colourswitcher}.
Find distinct $d_1,d_2,d_3,d_4\in [n]\setminus (C\cup \{c_1,c_2\})$ such that $d_1+d_2=d_3+d_4+k$ and $(1,c_1,d_1,d_2,-d_3,-d_4)$ and $(1,c_1+k,d_1,d_2,-d_3,-d_4)$ are both valid paths (that is, they have distinct vertices) starting at $1$. This is possible, as follows.
Pick $d_1,d_2,d_3\in [n]\setminus (C\cup \{c_1,c_2\})$ distinctly in turn so that $(1,c_1,d_1,d_2,-d_3)$ and $(1,c_1+k,d_1,d_2,-d_3)$ are both valid paths with the first one avoiding $1+c_1+k$ and the second one avoiding $1+c_2$. Note that, as we choose each $d_i$, we add one more vertex to each of these paths, which have at most 4 vertices already, so at most 12 colours are ruled out by the requirement these paths are valid and avoid the mentioned vertices. Furthermore, there are at most $|C|+4\leq n/20$ colours in $C\cup \{c_1,c_2\}$ or already chosen as some $d_j$, $j<i$. Thus, there are at least $n/2$ choices for each $d_i$, and therefore at least $(n/2)^3$ choices in total.
Now, let $d_4:=d_1+d_2-d_3-k$. Note that there are at most $n^2(|C|+5)\leq n^3/20$ choices for $\{d_1,d_2,d_3\}$ so that $d_4\in C\cup\{c_1,c_2,d_1,d_2,d_3\}$. Therefore, we can choose distinct $d_1,d_2,d_3\in [n]\setminus (C\cup \{c_1\cup c_2\})$ so that $d_4\notin C\cup\{c_1,c_2,d_1,d_2,d_3\}$, in addition to $(1,c_1,d_1,d_2,-d_3)$ and $(1,c_1+k,d_1,d_2,-d_3)$ both being valid paths which avoid $1+c_1+k$ and $1+c_2$, respectively. Noting that $1+c_1+d_1+d_2-d_3-d_4=1+c_1+k$, $(1,c_1,d_1,d_2,-d_3,-d_4)$ is therefore a valid path. Noting that $1+c_1+k+d_1+d_2-d_3-d_4=1+c_1+2k=1+c_2$, $(1,c_1+k,d_1,d_2,-d_3,-d_4)$ is therefore a valid path, with endvertex $1+c_2$. Therefore, its reverse, $(1+c_2,d_4,d_3,-d_2,-d_1,-c_1-k)$, is a valid path that ends with $1$. Moving the vertex $1$ to the start of the path, we get the valid path $(1,c_2,d_4,d_3,-d_2,-d_1)$.
Let $I$ be the set of $i\in [n]\setminus (C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\})$ such that $x$ and $y$ are not on $(x+i,c_1,d_1,d_2,-d_3,-d_4)$, or $(x+i,c_2,d_4,d_3,-d_2,-d_1)$, and that $y-i-c_1-k-x\notin C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\}$. Note that these conditions rule out at most 12, 12, and $|C|+6$ values of $i$ respectively, so that $|I|\geq n-2|C|-12-24\geq n/2$.
For each $i\in I$, add $x$ and $y$ as the start and end of
$(x+i,c_1,d_1,d_2,-d_3,-d_4)$ respectively, giving, as $k=d_1+d_2-d_3-d_4$ the path
\[
P_i:=(x,i,c_1,d_1,d_2,-d_3,-d_4,y-i-c_1-k-x).
\]
Then, add $x$ and $y$ as the start and end of
$(x+i,c_2,d_4,d_3,-d_2,-d_1)$ respectively, giving, as $d_4+d_3-d_1-d_2=-k=c_1-c_2+k$ the path
\[
Q_i:=(x,i,c_2,d_4,d_3,-d_2,-d_1,y-i-c_1-k-x).
\]
Let $C_i=\{i,d_1,d_2,d_3,d_4,y-i-c_1-k\}$. Note that, $P_i$ and $Q_i$ are both rainbow $x,y$-paths with length 7, with colour sets $(C_i\cup\{c_1\})$ and $(C_i\cup\{c_1\})$ respectively (indeed, when we add $-d_i, 1 \leq i \leq 4$ to get the next vertex of the path the colour of this edge is $d_i$ and ``$-$" just indicates in which direction we are moving).
Each vertex in $X$ can appear as the interior vertex of at most 6 different paths $P_i$ and 6 different paths $Q_i$. As $|I|\geq n/2 > 12|X|$, there must be some $j\in I$ for which the interior vertices of $P_j$ and $Q_j$ avoid $X$. Then, $C'=C_j$ is a colour set as required by the lemma, as demonstrated by the paths $P_j$ and $Q_j$.
\epsilonilonnd{proof}
The following lemma uses this to find colour switchers for an arbitary set of 100 colours $\{c_1,\ldots,c_{100}\}$ in the ND-colouring between an arbitrary vertex pair $\{x,y\}$. A sketch of its proof is as follows. First, we select a vertex $x_1$ and colours $d_1,\ldots,d_{100}$ so that $c_i+d_i=x_1-x$ for each $i\in [100]$. By choosing the vertex between $x$ and $x_1$ appropriately, we can find a $\{c_i,d_i\}$-rainbow $x,x_1$-path with length 2 for each $i\in [100]$. This allows us to use any pairs of colours $\{c_i,d_i\}$, so we need only construct a path which can switch between using any set of 99 colours from $\{d_1,\ldots,d_{100}\}$. This we do by constructing a sequence of $(d_i,d_{i+1})$-switchers for each $i\in [99]$ and putting them together between $x_1$ and $y$.
\begin{lemma}[1 in 100 colour absorbers]\label{lem-absorbpath}
Let $1\gg n^{-1}$. Suppose $K_{2n+1}$ is ND-coloured.
Suppose we have a pair of distinct vertices $x,y\in V(K_{2n+1})$, a set $C\subseteq C(K_{2n+1})$ of $100$ colours, and sets $X\subseteq V(K_{2n+1})$ and $C'\subseteq C(K_{2n+1})$ with $|X|\leq n/10^3$ and $|C'|\leq n/10^3$.
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Let $I$ be the set of $i\in [n]\setminus (C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\})$ such that $x$ and $y$ are not on $(x+i,c_1,d_1,d_2,-d_3,-d_4)$, or $(x+i,c_2,d_4,d_3,-d_2,-d_1)$, and that $y-i-c_1-k-x\notin C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\}$. Note that these conditions rule out at most 12, 12, and $|C|+6$ values of $i$ respectively, so that $|I|\geq n-2|C|-12-24\geq n/2$.
For each $i\in I$, add $x$ and $y$ as the start and end of
$(x+i,c_1,d_1,d_2,-d_3,-d_4)$ respectively, giving, as $k=d_1+d_2-d_3-d_4$ the path
\[
P_i:=(x,i,c_1,d_1,d_2,-d_3,-d_4,y-i-c_1-k-x).
\]
Then, add $x$ and $y$ as the start and end of
$(x+i,c_2,d_4,d_3,-d_2,-d_1)$ respectively, giving, as $d_4+d_3-d_1-d_2=-k=c_1-c_2+k$ the path
\[
Q_i:=(x,i,c_2,d_4,d_3,-d_2,-d_1,y-i-c_1-k-x).
\]
Let $C_i=\{i,d_1,d_2,d_3,d_4,y-i-c_1-k\}$. Note that, $P_i$ and $Q_i$ are both rainbow $x,y$-paths with length 7, with colour sets $(C_i\cup\{c_1\})$ and $(C_i\cup\{c_1\})$ respectively (indeed, when we add $-d_i, 1 \leq i \leq 4$ to get the next vertex of the path the colour of this edge is $d_i$ and ``$-$" just indicates in which direction we are moving).
Each vertex in $X$ can appear as the interior vertex of at most 6 different paths $P_i$ and 6 different paths $Q_i$. As $|I|\geq n/2 > 12|X|$, there must be some $j\in I$ for which the interior vertices of $P_j$ and $Q_j$ avoid $X$. Then, $C'=C_j$ is a colour set as required by the lemma, as demonstrated by the paths $P_j$ and $Q_j$.
\epsilonilonnd{proof}
The following lemma uses this to find colour switchers for an arbitary set of 100 colours $\{c_1,\ldots,c_{100}\}$ in the ND-colouring between an arbitrary vertex pair $\{x,y\}$. A sketch of its proof is as follows. First, we select a vertex $x_1$ and colours $d_1,\ldots,d_{100}$ so that $c_i+d_i=x_1-x$ for each $i\in [100]$. By choosing the vertex between $x$ and $x_1$ appropriately, we can find a $\{c_i,d_i\}$-rainbow $x,x_1$-path with length 2 for each $i\in [100]$. This allows us to use any pairs of colours $\{c_i,d_i\}$, so we need only construct a path which can switch between using any set of 99 colours from $\{d_1,\ldots,d_{100}\}$. This we do by constructing a sequence of $(d_i,d_{i+1})$-switchers for each $i\in [99]$ and putting them together between $x_1$ and $y$.
\begin{lemma}[1 in 100 colour absorbers]\label{lem-absorbpath}
Let $1\gg n^{-1}$. Suppose $K_{2n+1}$ is ND-coloured.
Suppose we have a pair of distinct vertices $x,y\in V(K_{2n+1})$, a set $C\subseteq C(K_{2n+1})$ of $100$ colours, and sets $X\subseteq V(K_{2n+1})$ and $C'\subseteq C(K_{2n+1})$ with $|X|\leq n/10^3$ and $|C'|\leq n/10^3$.
Then, there is a set $\bar{C}\subseteq C(K_{2n+1})\setminus (C\cup C')$ of 694 colours and a set $\bar{X}\subseteq V(K_{2n+1})\setminus X$ of at most $1500$ vertices so that, for each $c\in C$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $\bar{X}$.
\epsilonilonnd{lemma}
\begin{proof}
Let $\epsilonilonll=100$, $x_0=x$, and $x_\epsilonilonll=y$, and label $C=\{c_1,\ldots,c_\epsilonilonll\}$. Pick $x_1\in [2n+1]\setminus (X\cup\{x_0,x_\epsilonilonll\})$ so that $x_1-x_0+c_i\in [n]\setminus (C\cup C')$ and $x_1-c_i\in [2n+1]\setminus X$
for each $i\in [\epsilonilonll]$. Each such condition forbids at most $n/10^3+100$ points and we have at most $2\epsilonilonll=200$ conditions so we can indeed find $x_1$ satisfying all of them.
For each $i\in [\epsilonilonll]$, let $d_i=x_1-x_0+c_i$ and $y_i= x_1-c_i=x_0+d_i$. Then,
\begin{itemize}
\item $c_1,\ldots,c_\epsilonilonll,d_1,\ldots,d_\epsilonilonll$ are distinct colours in $C\cup ([n]\setminus C')$,
\item $\{x_0,x_1,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}$ are distinct vertices in $\{x,y\}\cup ([2n+1]\setminus X)$, and
\item for each $i\in [\epsilonilonll]$, $x_0y_ix_1$ is a $\{c_i,d_i\}$-rainbow path.
\epsilonilonnd{itemize}
Let $C''=\{d_1,\ldots,d_\epsilonilonll\}$.
Pick distinct vertices $x_2,\ldots,x_{\epsilonilonll-1}\in [2n+1]\setminus (X\cup\{x_0,x_1,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\})$ and let $X'=\{x_0,x_1,\ldots,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}$.
Next, iteratively, for each $1\leq i\leq \epsilonilonll-1$, using Lemma~\ref{lem-switchpath} find a set $X_i$ of at most 12 vertices in $[2n+1]\setminus (X\cup X'\cup(\cup_{j<i}X_j))$ and a set $C_i$ of 6 colours in $[n]\setminus (C\cup C'\cup C''\cup(\cup_{j<i}C_j))$ so that there is a $(C_i\cup\{d_{i}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$ and a $(C_i\cup\{d_{i+1}\})$-rainbow
$x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$.
Let $\bar{C}=C''\cup(\cup_{i\in [\epsilonilonll-1]}C_i)$ and $\bar{X}=X'\cup (\cup_{i\in [\epsilonilonll-1]}X_i)$, and note that $|\bar{C}|=\epsilonilonll+6(\epsilonilonll-1)=694$ and
$|\bar{X}|\leq 2\epsilonilonll+1+12(\epsilonilonll-1) \leq 1500$. We will show that $\bar{C}$ and $\bar{X}$ satisfy the condition in the lemma.
Let then $j\in [\epsilonilonll]$. For each $1\leq i< j$, let $P_i$ be a $(C_i\cup\{d_{i}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$. For each $j\leq i\leq \epsilonilonll-1$, let $P_i$ be a $(C_i\cup\{d_{i+1}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$. Thus, the paths $P_i$, $i\in [\epsilonilonll-1]$, cover all the colours in $C''$ except for $d_j$, as well as the colours in each set $C_i$.
Therefore, as $x_0=x$ and $x_\epsilonilonll=y$,
\[
x_0y_jx_1P_1x_2P_2x_3P_3\ldots P_{\epsilonilonll-1}x_{\epsilonilonll}
\]
is a $(\bar{C}\cup \{c_j\})$-rainbow $x,y$-path with length $2+7\times 99=695$ whose interior vertices are all in $\bar{X}$, as required.
\epsilonilonnd{proof}
The following corollary of this will be convenient to apply.
\begin{corollary}\label{cor-absorbpath}
Let $1\gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. For each pair of distinct vertices $x,y\in V(K_{2n+1})$ and each set $C\subseteq C(K_{2n+1})$ of 100 colours, there are $\epsilonilonll=n/10^7$ disjoint vertex sets $X_1,\ldots,X_\epsilonilonll\subseteq V(K_{2n+1})\setminus \{x,y\}$ with size at most 1500 and disjoint colour sets $C_1,\ldots,C_\epsilonilonll\subseteq [n]\setminus C$ with size 694
such that the following holds.
For each $i\in [\epsilonilonll]$ and $c\in C$, there is a $C_i\cup\{c\}$-rainbow $x,y$-path with length 695 and interior vertices in $X_i$.
\epsilonilonnd{corollary}
\begin{proof}
Iteratively, for each $i=1, \dots, \epsilonilonll$, choose sets $X_i$ and $C_i$ using Lemma~\ref{lem-absorbpath} (at the $i$th iteration letting $X=X_1\cup \dots \cup X_{i-1}$, $C'=C_1\cup \dots \cup C_{i-1}$).
\epsilonilonnd{proof}
The following lemma finds $1$-in-$100$ colour switchers in a random set of colours and vertices.
\begin{lemma}[Colour switchers using random vertices and colours]\label{lem-randabsorbpath}
Let $p,q\gg \mu \gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. Let $X\subseteq V_{2k+1}$ be $p$-random and $C\subseteq C(K_{2n+1})$ $q$-random, and such that $X$ and $C$ are independent. With high probability the following holds.
For every distinct $x,y\in V(K_{2n+1})$, $C'\subseteq C(K_{2n+1})$ with $|C'|=100$, and $X'\subseteq X$, $C''\subseteq C$ with $|X'|,|C''|\leq\mu n$, there is a set $\bar{C}\subseteq C\setminus (C'\cup C'')$ of 694 colours and a set $X''\subseteq X\setminus X'$ of at most $1500$ vertices with the following property. For each $c\in C'$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $X''$.
\epsilonilonnd{lemma}
\begin{proof}
We will show that for any distinct pair $x,y\in V(K_{2n+1})$ of vertices and set $C'\subseteq C(K_{2n+1})$ of 100 colours the property holds with probability $1-o(n^{-102})$ so that the lemma holds by a union bound.
Fix then distinct $x,y\in V(K_{2n+1})$ and a set $C'\subseteq C(K_{2n+1})$ with size 100. Fix $\epsilonilonll= n/10^7$ and use Corollary~\ref{cor-absorbpath} to find disjoint vertex sets $X_i\subseteq V(K_{2n+1})$, $i\in [\epsilonilonll]$, with size at most 1500 and disjoint colours sets $C_i\subseteq C(K_{2n+1})$, $i\in [\epsilonilonll]$, with size $694$ so that for each $c\in C'$ and $i\in [\epsilonilonll]$ there is a $C_i\cup \{c\}$-rainbow $x,y$-path with length 695 and internal vertices in $X_i$.
Let $I\subseteq [\epsilonilonll]$ be the set of $i\in [\epsilonilonll]$ for which $X_i\subseteq X$ and $C_i\subseteq C$. Note that $|I|$ is $1$-Lipschitz, and, for each $i\in [\epsilonilonll]$, we have $\mathbb{P}(X_i\subseteqeq X, C_i\subseteqeq C)\geq p^{1500}q^{695}\gg \mu$.
By Azuma's inequality, with probability $1-o(n^{-102})$ we have $|I|\geq 10^4\mu n$.
Take then any $X'\subseteq X$ and $C''\subseteq C$ with size at most $\mu n$ each. There must be some $j\in I$ for which $X'\cap X_j=\epsilonilonmptyset$ and $C''\cap C_j=\epsilonilonmptyset$. Let $\bar{C}=C_j\subseteq C\setminus (C'\cup C'')$ and $X''=X_j\subseteq X\setminus X'$. Then, as required, for each $c\in C'$ there is a $(\bar{C}\cup\{c\})$-rainbow $x,y$-path with interior vertices in $X''$.
\epsilonilonnd{proof}
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\subsection{Distributive absorption with paths}
We now use Lemma~\ref{lem-randabsorbpath} and distributive absorption to get a larger scale absorption property, as follows.
\begin{lemma}\label{absorbBmacro} Let $1/n\ll \epsilonilonta \ll \mu\ll \epsilonilonps$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0$ is an $\epsilonilonps$-random subset of $V(K_{2n+1})$ and $C_0$ is an $\epsilonilonps$-random subset of $C(K_{2n+1})$, which is independent of $V_0$. Suppose $\epsilonilonll=\mu n/695$ and that $\epsilonilonll/3\in \mathbb{N}$. With high probability, the following holds.
Suppose that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})\setminus C_0$ is a set of at most $\epsilonilonta n$ colours. Then, there is a set $\hat{C}\subseteq C_0$ of $695\epsilonilonll-\alpha n$ colours such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C')$-rainbow, have length 695, and internal vertices in $V_0$.
\epsilonilonnd{lemma}
\begin{proof} By Lemma~\ref{lem-randabsorbpath} (applied to $X=V_0, C=C_0$ with $p=q=\epsilonilonps$ and $\mu'=3\mu$), we have the following property with high probability.
\stepcounter{propcounter}
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For each distinct $x,y\in V(K_{2n+1})$, and every $C'\subseteq C(K_{2n+1})$ with $|C'|=100$, and $X'\subseteq V_0$, $C''\subseteq C_0$ with $|X'|,|C''|\leq 3\mu n$, there is a set $\bar{C}\subseteq C_0\setminus (C'\cup C'')$ of 694 colours and a set $X''\subseteq V_0\setminus X'$ of at most $1500$ vertices with the following property.
For each $c\in C'$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $X''$.\label{bbb2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Suppose then that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colors.
Let $h=\epsilonilonll/3$. By Lemma~\ref{Lemma_Chernoff}, with high probability
$|C_0|\geq \epsilonilonpsilon n/2> \mu n$. Therefore we can pick a set $\hat C_0 \subseteq C_0$ of $(3h-\alpha n)$ colours. Noting that $|\hat C_0|\geq 2h$ and $|\hat C_0\cup C|\leq 4h$.
By the same reasoning as just before \ref{Hmatch}, by Lemma~\ref{Lemma_H_graph} there is a bipartite graph, $H$ with maximum degree 100 say, with vertex classes $[3h]$ and $\hat C_0\cup C$ such that the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For each set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect matching between $[3h]$ and $\hat C_0\cup C'$ in $H$.\label{Hmatch2}
\epsilonilonnd{enumerate}
Iteratively, for each $1\leq i\leq 3h$, let $D_i=N_H(i)$ and, using \ref{bbb2}, find sets $C_i\subseteq C_0\setminus (C_1\cup\ldots\cup C_{i-1})$ and $V_i\subseteq V_0\setminus (V_1\cup \ldots\cup V_{i-1})$,
with sizes $694$ and at most $1500$, such that the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For each $c\in D_i$, there is a $(C_i\cup\{c\})$-rainbow $x_i,y_i$-path with length 695 with internal vertices in $V_i$.\label{propabsorb2}
\epsilonilonnd{enumerate}
Note that in the $i$th application of \ref{bbb2}, we have $X'=V_1\cup \ldots\cup V_{i-1}$ and $C''=C_1\cup\ldots \cup C_{i-1}$, so that $|X'|\leq 1500\epsilonilonll\leq 3\mu n$ and $|C'|\leq 694\epsilonilonll\leq 3\mu n$, as required.
Let $\hat{C}=\hat C_0\cup C_1\cup \ldots\cup C_{3h}$. We claim $\hat{C}$ has the property required. Indeed, suppose $C'\subseteq C$ is a set of $\alpha n$
colours. Using~\ref{Hmatch2}, let $M$ be a perfect matching between $[3h]$ and $\hat{C}_0\cup C'$ in $H$, and label $\hat C_0\cup C'=\{c_1,\ldots,c_{3h}\}$ so that, for each $i\in [3h]$, $c_i$ is matched to $i$ in $M$.
By \ref{propabsorb2}, for each $i\in [3h]$, there is an $x_i,y_i$-path, $P_i$ say, with length 695 which is $(C_i\cup\{c\})$-rainbow with internal vertices in $V_i$. Then, $P_1,\ldots, P_{3h}$ are the paths required.
\epsilonilonnd{proof}
The following corollary of Lemma~\ref{absorbBmacro} will be convenient to apply.
\begin{corollary}\label{cor-absorbBmacro} Let $1/n\ll \epsilonilonta \ll \mu\ll \epsilonilonps$ and $1/k\ll 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0$ is an $\epsilonilonps$-random subset of $V(K_{2n+1})$ and $C_0$ is an $\epsilonilonps$-random subset of $C(K_{2n+1})$, which is independent of $V_0$. Suppose that $\epsilonilonll=\mu n/k$ and $695|k$. With high probability, the following holds.
Suppose that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colours. Then, there is a set $\hat{C}\subseteqeq C_0$ of $k\epsilonilonll-\alpha n$ colours such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C')$-rainbow, have length k, and internal vertices in $V_0$.
\epsilonilonnd{corollary}
\begin{proof} Let $V_1,V_2\subseteq V_0$ be disjoint and $(\epsilonilonps/2)$-random and let $C_0'\subseteq C_0$ be $(\epsilonilonps/2)$-random. Let $\epsilonilonll'=\mu n/695=\epsilonilonll \cdot k/695$. We can assume that $3|\epsilonilonll'$, as discussed in Section~\ref{sec:not}.
By Lemma~\ref{absorbBmacro} (applied with $\epsilonilonps'=\epsilonilonps/2$, $\mu=\mu$, $\epsilonilonta=\epsilonilonta$, $C_0'=C_0'$ and $V_0'=V_1$) and Lemma~\ref{Lemma_Chernoff}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item
Let $\{x_1,\ldots,x_{\epsilonilonll'},y_1,\ldots,y_{\epsilonilonll'}\}\subseteq V(K_{2n+1})$, $\alpha'\leq \epsilonilonta$ and let $C'\subseteq C(K_{2n+1})$ be a set of at most $\epsilonilonta n$ vertices. Then, there is a set $\hat{C}$ of $695\epsilonilonll'-\alpha' n$ colours such that, for every set $C''\subseteq C$ of $\alpha' n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C'')$-rainbow, have length 695, and internal vertices in $V_1$.\label{argh1}
\item $|V_2|\geq \mu n/4$.\label{argh2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Suppose therefore that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colours. Let $k'=k/695\in \mathbb{N}$ and, let $\{z_{i,j}:i\in [\epsilonilonll],j\in [k']\}$ be a set of vertices in $V_2$, using \ref{argh2}. Apply \ref{argh1} to the pairs $(x_i,z_{i,1})$, $(z_{i,j},z_{i,j+1})$ and $(z_{i,k'},y_i)$, $i\in [\epsilonilonll]$, $1\leq j<k'$, and take their union, to get the required paths.
\epsilonilonnd{proof}
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\subsection{Covering small colour sets with paths}
We now show that paths can be found using every colour in an arbitrary small set of colours.
\begin{lemma}\label{colourcoverB} Let $1/n\ll \xi \ll\beta$, let $1/n\ll 1/k\ll 1$ with $k= 3 \mod 4$.
Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $C_0\subseteq C(K_n)$ are $\beta$-random and independent of each other. With high probability, the following holds with $m=5\xi n/k$.
For any set $C\subseteq C(K_{n+1})\setminus C_0$ of at most $\xi n$ colours, and any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus V_0$, there is a set of vertex-disjoint paths $x_i,y_i$-paths, $i\in [m]$, each with length $k$ and interior vertices in $V_0$, which are collectively $(C\cup C_0)$-rainbow and use all the colours in $C$.
\epsilonilonnd{lemma}
\begin{proof} Let $V_1,V_2\subseteq V_0$ be disjoint and $(\beta/2)$-random. Let $C_1,C_2\subseteq C_0$ be disjoint and $(\beta/2)$-random.
By Lemma~\ref{Lemma_Chernoff}, and Lemma~\ref{Lemma_number_of_colours_inside_random_set}, with high probability, the following properties hold.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $|C_1|\geq \beta n/3$.\label{beep1}
\item $V_1$ is $(\beta^2 n/4)$-replete. \label{beep1prime}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_few_connecting_paths} applied with $p=\beta /2$, $q=4\xi$, $V=V_2$ and $C=C_2$, with high probability, we have the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any $m'\leq 4\xi n$ and set $\{x_1, y_1, \dots, x_{m'}, y_{m'}\}\subseteq V(K_{2n+1})$
there is a collection $P_1, \dots, P_{m'}$ of vertex-disjoint paths with length $3$, with internal vertices in $V_2$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [m']$, and $P_1\cup \dots\cup P_{m'}$ is $C_2$-rainbow.\label{beep2}
\epsilonilonnd{enumerate}
We will now show that the property in the lemma holds. Let then $C\subseteq C(K_{n+1})\setminus C_0$ have at most $\xi n$ colours and let $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus V_0$. Take $\epsilonilonll=(k-3)/4$, and note that this is an integer and $\epsilonilonll m\geq \xi n$. Using \ref{beep1}, take an order $\epsilonilonll m$ set $C'\subseteq C\cup C_1$ with $C\subseteq C'$ and label $C'=\{c_{i,j}:i\in [m],j\in [\epsilonilonll]\}$.
Using \ref{beep1prime}, greedily find independent edges $s_{i,j}t_{i,j}$, $i\in [m],j\in [\epsilonilonll]$, with vertices in $V_1$, so that each edge $s_{i,j}t_{i,j}$ has colour $c_{i,j}$. Note that, when the edge $s_{i,j}t_{i,j}$ is chosen at most $2m\epsilonilonll\leq 2mk\leq 10\xi n$ vertices in $V_1$ are in already chosen edges.
Using \ref{beep2} with $m'=m(\epsilonilonll+1)$, find vertex-disjoint paths $P_{i,j}$, $i\in [m]$, $0\leq j\leq \epsilonilonll$, with length 3 and internal vertices in $V_2$ so that these paths are collectively $C_2$-rainbow and the following holds for each $i\in [m]$.
\begin{itemize}
\item $P_{i,0}$ is a $x_{i}s_{i,1}$-path.
\item For each $1\leq j<\epsilonilonll$, $P_{i,j}$ is a $t_{i,j},s_{i,j+1}$-path.
\item $P_{i,\epsilonilonll}$ is a $t_{i,\epsilonilonll},y_i$-path.
\epsilonilonnd{itemize}
Then, the paths $P_i=\cup_{0\leq j\leq \epsilonilonll}P_{i,j}$, $i\in [m]$, use each colour in $C'\subseteq C\cup C_1$, and hence $C$, and otherwise use colours in $C_2$, have interior vertices in $V_0$ and, for each $i\in [m]$, $P_i$ is a length $3(\epsilonilonll+1)+\epsilonilonll=k$ path from $x_i$ to $y_i$. That is, the paths $P_i$, $i\in[m]$, satisfy the condition in the lemma. \epsilonilonnd{proof}
\subsection{Almost-covering colour sets with paths}
We now show that paths can be found using almost every colour in a set of mostly-random colours.
\begin{lemma}\label{finishingB}
Let $1\geq p\gg q\gg \gamma,\epsilonilonta\gg 1/k \gg 1/n$ with $k=1\mod 3$, and let $m\leq 1.01q n/k$.
Let $K_{2n+1}$ be $2$-factorized.
Suppose we have disjoint sets $V, V_0\subseteqeq V(K_{2n+1})$, and $D,D_0\subseteqeq C(K_{2n+1})$ with $V$ $p$-random, $D$ $q$-random, and $V_0, D_0$ $\gamma$-random. Suppose further that $V_0$ and $D_0$ are independent of each other. Then, the following holds with high probability.
For any set $C\subseteq C(K_{n+1})$ with $D\cup D_0\subseteq C$ of $mk+\epsilonilonta n$ colours, and any set $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$, there is a set of vertex-disjoint paths $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof} Let $\epsilonilonll=(k-4)/3$ and note that this is an integer. Let $m'=m+\epsilonilonta n/6k$. Using that $p\gg q,\epsilonilonta$, take in $V$ vertex disjoint sets $V',V_1,\ldots,V_{\epsilonilonll}$, such that $V'$ is $(p/2)$-random and, for each $i\in [\epsilonilonll]$, $V_i$ is $(m'/2n)$-random. Take an $(\epsilonilonta\gamma)$-random subset $D'_0\subseteq D_0$. Take in $D$ vertex disjoint sets $C_1,\ldots,C_{2\epsilonilonll}$ so that, and, for each $1\leq i\leq 2\epsilonilonll$, $C_i$ is $(m'/n)$-random.
Note that this later division is possible as $2\epsilonilonll\cdot m'/n\leq 2\epsilonilonll \cdot 1.02q n/k\leq q$.
By Lemma~\ref{Lemma_number_of_colours_inside_random_set}, with high probability the following holds.
\addtocounter{propcounter}{1}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $V'$ is $(p^2 n/4)$-replete.\label{mice1}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_few_connecting_paths}, applied with $p'=\gamma\epsilonilonta$ and $q'=2q/k$ to $D'_0$ and a $(\gamma\epsilonilonta)$-random subset of $V_0$, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $\{x_1, y_1, \dots, x_{2m}, y_{2m}\}\subseteq V(K_{2n+1})$
there is a collection $P_1, \dots, P_{2m}$ of vertex-disjoint paths with length $3$, having internal vertices in $V_0$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [2m]$, and $P_1\cup \dots\cup P_{2m}$ is $D'_0$-rainbow.\label{mice1a}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_Chernoff}, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $|D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll}|\leq 2m'\epsilonilonll +\epsilonilonta n/2$. \label{mice1b}
\item For each $j\in [\epsilonilonll]$, $|V_j|\leq m'+\epsilonilonta^2 n/k^2$.\label{mice1bb}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_nearly_perfect_matching}, applied for each $i\in [2\epsilonilonll]$ and $j\in [\epsilonilonll]$ with $p'=m'/n$ and $\beta=\epsilonilonta^2/k^2$, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{5}
\item For each $i\in [2\epsilonilonll]$ and $j\in [\epsilonilonll]$ and any vertex set $Y\subseteq V(K_{2n+1})\setminus V$ with $|Y|\leq m'$ there is a $C_i$-rainbow matching from $Y$ into $V_j$ with at least $|Y|-\epsilonilonta^2 n/k^2$ edges.\label{mice2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Set then $C\subseteq C(K_{n+1})$ with $D\cup D_0\subseteq C$ so that $|C|=mk+\epsilonilonta n$ and let $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V_0\cup V)$.
Note that, using \ref{mice1b},
\begin{align*}
|C\setminus (D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll})|&\geq mk+\epsilonilonta n-2m'\epsilonilonll-\epsilonilonta n/2=
(3\epsilonilonll+4)m+\epsilonilonta n/2-2m'\epsilonilonll\geq 3\epsilonilonll (m-m')+\epsilonilonta n/2+m'\epsilonilonll\\
&= -3\epsilonilonll \epsilonilonta n/6k+\epsilonilonta n/2+m'\epsilonilonll
\geq m'\epsilonilonll,
\epsilonilonnd{align*}
and take disjoint sets $C'_1,\ldots,C_\epsilonilonll'\subseteq C\setminus (D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll})$ with size $m'$.
Let $C'=C'_1\cup\ldots \cup C'_\epsilonilonll$. Greedily, using~\ref{mice1}, take vertex disjoint edges $x_cy_c$, $c\in C'$, with vertices in $V'$ so that the edge $x_cy_c$ has colour $c$. Note that these edges have $2m'\epsilonilonll \leq qn$ vertices in total, so that this greedy selection is possible.
Let $M$ be the matching $\{x_cy_c:c\in C'\}$.
For each $i\in [\epsilonilonll]$, let $Z_i=\{x_c:c\in C'_i\}$ and $Y_i=\{y_c:c\in C'_i\}$.
For each $i\in [\epsilonilonll-1]$, use~\ref{mice2} to find a $C_{i}$-rainbow matching $M_i$ with $m'-\epsilonilonta^2 n/k^2$ edges from $Z_i$ into $V_i$, and a $C_{i+\epsilonilonll}$-rainbow matching, $M'_i$ say, with $m'-\epsilonilonta^2 n/k^2$ edges from $Y_i$ into $V_{i-1}$. Note that, by \ref{mice1bb} these matchings overlap in at least $m'-4\epsilonilonta^2 n/k^2$ vertices.
Therefore, putting together $M$ with the matchings $M_i$, $M_i'$, $i\in [\epsilonilonll]$ gives at least $m'-\epsilonilonll \cdot 4\epsilonilonta^2 n/k^2\geq m$ vertex disjoint paths with length $3\epsilonilonll-2$. Furthermore, these paths are collectively rainbow with colours in $(C\setminus D'_0)$. Take $m$ such paths, $Q_i$, $i\in [m]$. Apply \ref{mice1a}, to connect one endpoint of $Q_i$ to $x_i$ and another endpoint to $y_i$ using two paths of length 3 and new vertices and colours in $V_0$ and $D'_0$ respectively to get the paths with length $k=3\epsilonilonll+4$ as required.
\epsilonilonnd{proof}
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\subsection{Proof of the finishing lemma in Case B}\label{subsec:finishB}
We can now prove Lemma~\ref{lem:finishB}.
\begin{proof}[Proof of Lemma~\ref{lem:finishB}]
Pick $\xi,\beta,\lambda,\alpha$ so that $1/k\ll \xi\ll\beta\ll\lambda\ll \alpha\ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll p\leq 1$. Let $V_1',V_2',V_3'\subseteq V$ be disjoint sets which are each $(p/3)$-random. Let $W_1,W_2,W_3\subseteq V_0$ be disjoint sets which are each $(\mu/3)$-random.
Let $D_1,D_2,D_3\subseteq C_0$ be disjoint and $(\mu/3)$-, $\beta$- and $\alpha$-random respectively.
Set $m_1=(\epsilonilonps-5\xi-\lambda)n/k$, $m_2=5\xi n/k$ and $m_3=\lambda n/k$. By Lemma~\ref{Lemma_Chernoff}, with high probability we have that $|D_2|\leq 2\beta n$.
By Lemma~\ref{finishingB}, applied with $p'=p/3$, $q=(1-\epsilonilonta)\epsilonilonps$, $\gamma=\mu/3$, $\epsilonilonta'=\xi$, $n=n$, $k=k$, $m=m_1$, $V'=V_1'$, $V_0=W_1$, $D=C$, $D_0=D_1$, with high probability we have the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any set $\bar{C}\subseteq C(K_{n+1})$ with $C\cup D_1\subseteq \bar{C}$ of $m_1k+\xi n=(\epsilonilonps-4\xi-\lambda)n$ colours, and any collection of vertices $\{x_1,\ldots,x_{m_1},y_1,\ldots,y_{m_1}\}\subseteq V(K_{2n+1})\setminus (V_1'\cup W_1)$, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [m_1]$, each with length $k$, which have interior vertices in $V_1'\cup W_1$ and which are collectively $\bar{C}$-rainbow.
\label{mouseb11}
\epsilonilonnd{enumerate}
By Lemma~\ref{colourcoverB}, applied with $m=m_2$, $V_0\subseteq W_2$ a $\beta$-random subset and $C_0=D_2$, with high probability, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $\bar{C}\subseteq C(K_{n+1})\setminus D_2$ of at most $\xi n$ colours, and any set $\{x_1,\ldots,x_{m_2},y_1,\ldots,y_{m_2}\}\subseteq V(K_{2n+1})\setminus W_2$, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [{m_2}]$, each with length $k$ and interior vertices in $W_2$, which are collectively $(\bar{C}\cup D_2)$-rainbow and use all the colours in $\bar{C}$.\label{mouseb2}
\epsilonilonnd{enumerate}
By Corollary~\ref{cor-absorbBmacro}, applied with $\epsilonilonta'=2\beta$, $\mu'=\lambda$, $\epsilonilonps'=\alpha$, $n=n$, $k=k$, $\epsilonilonll=m_3$, $V_0\subseteq W_3$ an $\alpha$-random subset and $C_0=D_3$, with high probability we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any $\{x_1,\ldots,x_{m_3},y_1,\ldots,y_{m_3}\}$, $\bar{\beta}\leq 2\beta$ and $\bar{C}\subseteq C(K_{2n+1})$ with $|\bar{C}|\leq 2\beta n$, there is a set $D_3'\subseteq D_3$ of $m_3k-\bar{\beta}n$ colours
such that, for every set $C'\subseteq \bar{C}$ of $\bar{\beta} n$ colours, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [m_3]$, which are collectively $(D_3'\cup C')$-rainbow, have length $k$, and internal vertices in $W_3$.\label{mouseb3}
\epsilonilonnd{enumerate}
Let $m=\epsilonilonps n/k$, so that $m=m_1+m_2+m_3$. We will show that the property in the lemma holds.
Suppose then that $x_1,\ldots,x_m,y_1,\ldots,y_m$ are distinct vertices in $V(K_{2n+1})\setminus (V\cup V_0)$ and $D\subseteq C(K_{2n+1})$ so that $|D|=\epsilonilonps n$ and $C\cup C_0\subseteq D$. Let $V_1=V_1'\cup W_1$, $V_2=V_2'\cup W_2$ and $V_3=V_3'\cup W_3$.
Let $[m]=I_1\cup I_2\cup I_3$ be a partition with $|I_i|=m_i$ for each $i\in [3]$.
By \ref{mouseb3} (applied with $\bar C=D_2$ and $\bar{\beta}=(|D_2|/n)-4\xi$), there is a set $D_3'\subseteq D_3$ of $m_3k-|D_2|+4\xi n$ colours with the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item For every set $C'\subseteq D_2$ of $|D_2|-4\xi n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths, $i\in I_3$, which are collectively $(D_3'\cup C')$-rainbow, have length $k$, and internal vertices in $W_3$.\label{mouseb4}
\epsilonilonnd{enumerate}
Let $C_1=D\setminus (D_2\cup D_3')$, $C_2=D_2$ and $C_3=D_3'$. We now reason analogously to the discussion after \ref{propp1}--\ref{propp3}.
Note that $C\cup D_1\subseteq C_1$ and
\begin{equation}
|C_1|=|D|-|D_2|-(m_3k-|D_2|+4\xi n)=mk-m_3k-4\xi n=(m_1+m_2)k-4\xi n=m_1k+\xi n.
\label{coldrain}
\epsilonilonnd{equation}
Therefore, using \ref{mouseb11}, we can find $m_1$ paths $\{P_1, \dots, P_{m_1}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_1$, so that these paths are vertex disjoint with internal vertices in $V_1=V_1'\cup W_1$ and are collectively $C_1$-rainbow.
Let $C'=C_1\setminus (\cup_{i\in X_1}C(P_i))$, so that, by \epsilonilonqref{coldrain}, $|C'|=\xi n$.
Using \ref{mouseb2}, we can find $m_2$ paths $\{P_1, \dots, P_{m_2}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_2$, so that these paths are vertex disjoint with internal vertices in $V_2=V_2'\cup W_2$ and are collectively $(C_2\cup C')$-rainbow and use every colour in $C'$.
Let $C''=C_2\setminus (\cup_{i\in X_2}C(P_i))$, and note that $|C''|=|C_2\cup C'|-m_2k=|D_2|+\xi n-m_2k=|D_2|-4\xi n$.
Using \ref{mouseb4} and that $C_3=D_3'$, we can find $m_3$ paths $\{P_1, \dots, P_{m_3}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_3$, so that the paths are vertex disjoint with internal vertices in $V_3=V_3'\cup W_3$ which are collectively $(C_3\cup C'')$-rainbow.
Then, for each $i\in [\epsilonilonll]$, the path $P_i$ is an $x_i,y_i$-path with length $k$, so that all the paths are vertex disjoint with internal vertices in $V_1\cup V_2\cup V_3\subseteq V_0\cup V$ and which are collectively $D$-rainbow, as required.
\epsilonilonnd{proof}
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\section{Randomized tree embedding}\label{sec:almost}
In this section, we prove Theorem~\ref{nearembedagain}. We start by formalising what we mean by a random rainbow embedding of a tree.
\begin{definition}
For a probability space $\Omega$, tree $T$ and a coloured graph $G$, a randomized rainbow embedding of $T$ into $G$ is
a triple $\phi=(V_\phi, C_\phi, T_\phi)$ consisting of a random set of vertices $V_\phi:\Omega\to V(G)$, a random set of colours $C_\phi:\Omega\to C(G)$, and a random subgraph $T_\phi:\Omega\to E(G)$ such that:
\begin{itemize}
\item $V(T_\phi)\subseteqeq V_\phi$ and $C(T_\phi)\subseteqeq C_\phi$ always hold.
\item With high probability, $T_\phi$ is a rainbow copy of $T$.
\epsilonilonnd{itemize}
\epsilonilonnd{definition}
We will embed a tree bit by bit, starting with a randomized rainbow embedding of a small tree and then extending it gradually. For this, we need a concept of one randomized embedding \epsilonilonmph{extending} another.
\begin{definition}\label{Definition_embedding_extension}
Let $G$ be a graph and $T_1\subseteqeq T_2$ two nested trees. Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, S_{\phi_1})$ and $\phi_2=(V_{\phi_2}, C_{\phi_2}, S_{\phi_2})$ be randomized rainbow embeddings of $T_1$ and $T_2$ respectively into $G$. We say that $\phi_2$ extends $\phi_1$ if
\begin{itemize}
\item $T_{\phi_1}\subseteqeq T_{\phi_2}$, $C_{\phi_1}\subseteqeq C_{\phi_2}$, and $V_{\phi_1}\subseteqeq V_{\phi_2}$ always hold.
\item $V(T_{\phi_2})\setminus V(T_{\phi_1})\subseteq V_{\phi_2}\setminus V_{\phi_1}$ and $C(T_{\phi_2})\setminus C(T_{\phi_1})\subseteq C_{\phi_2}\setminus C_{\phi_1}$ always hold.
\epsilonilonnd{itemize}
\epsilonilonnd{definition}
The above definition implicitly assumes that the two randomized embeddings are defined on the same probability space, which is the case in our lemmas, except for Lemma~\ref{Lemma_extending_with_large_stars}. When $\phi_1$ and $\phi_2$ are defined on different probability spaces $\Omega_{\phi_1}$ and $\Omega_{\phi_2}$ respectively, we use the following definition. We say that an extension of $\phi_1$ is a measure preserving transformation $f:\Omega_{\phi_2}\to \Omega_{\phi_1}$ (i.e.\ $\mathbb{P}(f^{-1}(A))=\mathbb{P}(A)$ for any $A\subseteqeq \Omega_{\phi_1}$). We say that $\phi_2$ extends $\phi_1$ if ``for every $\omega\in \Omega_{\phi_2}$ we have $T_{\phi_1}( f(\omega))\subseteqeq T_{\phi_2}$, $C_{\phi_1}( f(\omega))\subseteqeq C_{\phi_2}$, $V_{\phi_1}( f(\omega))\subseteqeq V_{\phi_2}$, and $V(T_{\phi_2})\setminus V(T_{\phi_1}( f(\omega)))$ is
contained in $V_{\phi_2}\setminus V_{\phi_1}( f(\omega))$ and $C(T_{\phi_2})\setminus C(T_{\phi_1}(f(\omega)))$ is contained in
$C_{\phi_2}\setminus C_{\phi_1}( f(\omega))$''. Such a measure preserving transformation ensures that $\phi_1'=(T_{\phi_1}\circ f, C_{\phi_1}\circ f, V_{\phi_1}\circ f)$ is a randomized embedding of $T_1$ defined on $\Omega_{\phi_2}$ which is equivalent to $\phi_1$ (i.e.\ the probability of any outcomes $\phi_1$ and $\phi_1'$ are the same). It also ensures that $\phi_2$ is an extension of $\phi_1'$ as in Definition~\ref{Definition_embedding_extension}.
The following lemma extends randomized embeddings of trees by adding a large star forest. Recall that a
$p$-random subset of some finite set is formed by choosing every element of it independently
with probability $p$.
\begin{lemma}[Extending with a large star forest]\label{Lemma_extending_with_large_stars}
Let $p\gg \beta\gg \gamma \gg d^{-1}, n^{-1}$ and $\log^{-1} n\gg d^{-1}$.
Let $T_1\subseteqeq T_2$ be forests such that $T_2$ is formed by adding stars with $\geq d$ leaves to vertices of $T_1$. Let $K_{2n+1}$ be $2$-factorized and suppose that $|T_2|=(1-p)n$.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$ where $V_{\phi_1}$ and $C_{\phi_1}$ are both $\gamma$-random.
Then, $\phi_1$ can be extended into a randomized rainbow embedding $\phi_2=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2$ so that $V_{\phi_2}$ and $C_{\phi_2}$ are $(1-p+\beta)$-random sets (with $V_{\phi_2}$ and $C_{\phi_2}$ allowed to depend on each other).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ such that $\beta\gg \alpha \gg \gamma $. Let $\theta =|T_1|/n$ and $F=T_2\setminus T_1$, noting that $F$ is a star forest with degrees $\geq d$ and $e(F)=(1-p-\theta)n$. Let $I\subseteqeq V(T_1)$ be the vertices to which stars are added to get $T_2$ from $T_1$.
Let $\Omega$ be the probability space for $\phi_1$.
We call $\omega\in \Omega$ \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and if we have $|V_{\phi_1}(\omega)|, |C_{\phi_1}(\omega)|\leq 4\gamma n$.
Using Chernoff's bound and the fact that $\phi_1$ is a randomized embedding of $T_1$ we have that with high probability a random $\omega \in \Omega$ is succesful.
For every successful $\omega$, let $J^{\omega}$ be the copy of $I$ in $T_{\phi_1}(\omega)$.
We can apply Lemma~\ref{Lemma_randomized_star_forest} with $F=F$, $J=J^{\omega}$,
$V=V(K_{2n+1})\setminus V_{\phi_1}(\omega)$, $C=C(K_{2n+1})\setminus C_{\phi_1}(\omega)$, $p'=p+\theta, \alpha=\alpha, \gamma'=4\gamma, d=d$ and $n=n$. This gives a probability space $\Omega^\omega$, and a randomized subgraph $F^{\omega}$, and randomized sets
$U^\omega\subseteqeq V(K_{2n+1})\setminus (V_{\phi_1}(\omega)\cup V(F^{\omega})),$ $D^{\omega}\subseteqeq C(K_{2n+1})\setminus (C_{\phi_1}(\omega)\cup C(F^{\omega}))$ (so $F^{\omega}$, $U^\omega$, $D^{\omega}$ are functions from $\Omega^{\omega}$ to the families of subgraphs/subsets of vertices/sets of colours of $K_{2n+1}$ respectively). From Lemma~\ref{Lemma_randomized_star_forest} we know that, for each $\omega$, $F^{\omega}$ is with high probability a copy of $F$, and that $U^{\omega}$ and $D^{\omega}$ are $(1-\alpha)(p+\theta)$-random subsets of $V(K_{2n+1})\setminus V_{\phi_1}(\omega)$ and $C(K_{2n+1})\setminus C_{\phi_1}(\omega)$ respectively. Setting $T^{\omega}=F^{\omega}\cup T_{\phi_1}(\omega)$ gives a subgraph which is a copy of $T_2$ with high probability (for successful $\omega$).
For every unsuccessful $\omega$, set $T^{\omega}=T_{\phi_1}(\omega)$, and choose $U^\omega\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}(\omega),$ $D^{\omega}\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}(\omega)$ to be independent $(1-\alpha)(p+\theta)$-random subsets (in this case letting $\Omega^{\omega}$ be an arbitrary probability space on which such $U^{\omega}, D^{\omega}$ are defined).
Let $\Omega_2=\{(\omega, \omega'): \omega\in \Omega, \omega'\in \Omega^{\omega}\}$ and set $\mathbb{P}((\omega, \omega'))=\mathbb{P}(\omega)\mathbb{P}(\omega')$. Notice that $\Omega_2$ is a probability space.
Let $T_{\phi_2}$ be a random subgraph formed by choosing $(\omega, \omega')\in \Omega_2$, and setting $T_{\phi_2}= T^\omega(\omega')$. Similarly define $U=U^{\omega}(\omega')$ and $D=D^{\omega}(\omega')$. Notice that $V\setminus V_{\phi_1}$ is $(1-\gamma)$-random and that $U|V_{\phi_1}$ is a $(1-\alpha)(p+\theta)$-random subset of $V\setminus V_{\phi_1}$. By Lemma~\ref{Lemma_mixture_of_p_random_sets}, $U$ is a $(1-\alpha)(1-\gamma)(p+\theta)$-random subset of $V(K_{2n+1})$. Similarly, $D$ is a $(1-\alpha)(1-\gamma)(p+\theta)$-random subset of $C(K_{2n+1})$. Since $\beta\gg \alpha,\gamma$, we have $(1-\alpha)(1-\gamma)(p+\theta)\geq p-\beta$, and therefore can choose $(p-\beta)$-random subsets $U'$ and $D'$ such that $U'\subseteqeq U$ and $D'\subseteqeq D$.
Now $V_{\phi_2}:=V(K_{2n+1})\setminus U'$ and $C_{\phi_2}=C(K_{2n+1})\setminus D'$ are $(1-p+\beta)$-random sets of vertices/colours.
We also have that with high probability $T_{\phi_2}$ is a copy of $T_2$ since
$$\mathbb{P}(\text{$T_{\phi_2}(\omega, \omega')$ is not a copy of $T_2$})\leq \mathbb{P}(\text{$\omega$ unsuccessful})+\mathbb{P}(\text{$\omega$ successful and $T_{\omega}$ not a copy of $T_2$}).$$
Thus the required extension of $\phi_1$ is $\phi_{2}=(T_{\phi_2}, V_{\phi_2}, C_{\phi_2})$ together with the measure preserving transformation $f:\Omega_2\to \Omega$ with $f:(\omega, \omega')\to\omega$.
\epsilonilonnd{proof}
The following lemma extends randomized embeddings of trees by adding connecting paths.
\begin{lemma}[Extending with connecting paths]\label{Lemma_extending_with_connecting_paths}
Let $p\gg q\gg n^{-1}$.
Let $T_1\subseteqeq T_2$ be forests such that $T_2$ is formed by adding $qn$ paths of length $3$ connecting different components of $T_1$. Let $K_{2n+1}$ be $2$-factorized.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$ and let $U\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $p$-random, independent subsets. Let $V_{\phi_2}=V_{\phi_1}\cup U$ and $C_{\phi_2}=C_{\phi_1}\cup D$.
Then, $\phi_1$ can be extended into a randomized rainbow embedding $\phi_2=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2$.
\epsilonilonnd{lemma}
\begin{proof}
Let $\Omega$ be the probability space for $\phi_1$.
We say that $\omega\in \Omega$ is \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and the conclusion of Lemma~\ref{Lemma_few_connecting_paths} holds with $V=U$, $C=D$, $p=p$, $q=q$ and $n=n$. As $\phi_1$ is a randomized embedding of $T_1$, and by Lemma~\ref{Lemma_few_connecting_paths}, we have that, with high probability, $\omega$ is successful.
For each successful $\omega$, let $x_1^{\omega}, y_1^{\omega}, \dots, x_{qn}^{\omega}, y_{qn}^{\omega}$ be the vertices of $T_{\phi_1}(\omega)$ which need to be joined by paths of length $3$ to get a copy of $T_2$. From the conclusion of Lemma~\ref{Lemma_few_connecting_paths}, for each successful $\omega$, we can find paths $P_i^\omega$, $i\in [qn]$, so that the paths are vertex disjoint and collectively $D$-rainbow, and each path $P_i^\omega$ is an $x_i,y_i$-path with length 3 and internal vertices in $U$.
Letting $T_{\phi_2}=T_{\phi_1}(\omega)\cup P_1^{\omega}\cup \dots\cup P_{qn}^{\omega}$, $(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ gives the required randomized embedding of $T_2$.
\epsilonilonnd{proof}
The following lemma extends randomized embeddings of trees by adding a sequence of matchings of leaves.
\begin{lemma}[Extending with matchings]\label{Lemma_extending_with_matchings}
Let $\epsilonilonll^{-1},p, q\gg n^{-1}$.
Let $T_1\subseteqeq T_{\epsilonilonll}$ be forests such that $T_{\epsilonilonll}$ is formed by adding a sequence of $\epsilonilonll$ matchings of leaves to $T_1$. Let $K_{2n+1}$ be $2$-factorized. Suppose $|T_{\epsilonilonll}|-|T_1|\leq pn$.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$. Let $U_{ind}\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D_{ind}\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $q$-random, independent subsets. Let $U_{dep}\subseteqeq V(K_{2n+1})\setminus (V_{\phi_1}\cup U_{ind})$ be $p/2$-random, and $D_{dep}\subseteqeq C(K_{2n+1})\setminus(C_{\phi_1}\cup D_{ind})$ $p$-random (possibly depending on each other).
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The following lemma extends randomized embeddings of trees by adding connecting paths.
\begin{lemma}[Extending with connecting paths]\label{Lemma_extending_with_connecting_paths}
Let $p\gg q\gg n^{-1}$.
Let $T_1\subseteqeq T_2$ be forests such that $T_2$ is formed by adding $qn$ paths of length $3$ connecting different components of $T_1$. Let $K_{2n+1}$ be $2$-factorized.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$ and let $U\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $p$-random, independent subsets. Let $V_{\phi_2}=V_{\phi_1}\cup U$ and $C_{\phi_2}=C_{\phi_1}\cup D$.
Then, $\phi_1$ can be extended into a randomized rainbow embedding $\phi_2=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2$.
\epsilonilonnd{lemma}
\begin{proof}
Let $\Omega$ be the probability space for $\phi_1$.
We say that $\omega\in \Omega$ is \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and the conclusion of Lemma~\ref{Lemma_few_connecting_paths} holds with $V=U$, $C=D$, $p=p$, $q=q$ and $n=n$. As $\phi_1$ is a randomized embedding of $T_1$, and by Lemma~\ref{Lemma_few_connecting_paths}, we have that, with high probability, $\omega$ is successful.
For each successful $\omega$, let $x_1^{\omega}, y_1^{\omega}, \dots, x_{qn}^{\omega}, y_{qn}^{\omega}$ be the vertices of $T_{\phi_1}(\omega)$ which need to be joined by paths of length $3$ to get a copy of $T_2$. From the conclusion of Lemma~\ref{Lemma_few_connecting_paths}, for each successful $\omega$, we can find paths $P_i^\omega$, $i\in [qn]$, so that the paths are vertex disjoint and collectively $D$-rainbow, and each path $P_i^\omega$ is an $x_i,y_i$-path with length 3 and internal vertices in $U$.
Letting $T_{\phi_2}=T_{\phi_1}(\omega)\cup P_1^{\omega}\cup \dots\cup P_{qn}^{\omega}$, $(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ gives the required randomized embedding of $T_2$.
\epsilonilonnd{proof}
The following lemma extends randomized embeddings of trees by adding a sequence of matchings of leaves.
\begin{lemma}[Extending with matchings]\label{Lemma_extending_with_matchings}
Let $\epsilonilonll^{-1},p, q\gg n^{-1}$.
Let $T_1\subseteqeq T_{\epsilonilonll}$ be forests such that $T_{\epsilonilonll}$ is formed by adding a sequence of $\epsilonilonll$ matchings of leaves to $T_1$. Let $K_{2n+1}$ be $2$-factorized. Suppose $|T_{\epsilonilonll}|-|T_1|\leq pn$.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$. Let $U_{ind}\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D_{ind}\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $q$-random, independent subsets. Let $U_{dep}\subseteqeq V(K_{2n+1})\setminus (V_{\phi_1}\cup U_{ind})$ be $p/2$-random, and $D_{dep}\subseteqeq C(K_{2n+1})\setminus(C_{\phi_1}\cup D_{ind})$ $p$-random (possibly depending on each other).
Then $\phi_1$ can be extended into a randomized rainbow embedding $\phi_{\epsilonilonll}=(V_{\phi_{\epsilonilonll}}, C_{\phi_{\epsilonilonll}}, T_{\phi_{\epsilonilonll}})$ of $T_{\epsilonilonll}$ into $V_{\phi_{\epsilonilonll}}=V_{\phi_1}\cup U_{ind}\cup U_{dep}$ and $C_{\phi_{\epsilonilonll}}=C_{\phi_1}\cup D_{ind}\cup D_{dep}$.
\epsilonilonnd{lemma}
\begin{proof}
Without loss of generality, suppose that $|T_{\epsilonilonll}|-|T_1|= pn$.
Define forests $T_1, \dots, T_{\epsilonilonll}$, and $p_1, \dots, p_{\epsilonilonll}\in [0,1]$ such that each $T_{i+1}$ is constructed from $T_i$ by adding a matching of $p_in$ leaves.
Randomly partition $U_{dep}=\{U_{dep}^1, \dots, U_{dep}^{\epsilonilonll}\}$ and $D_{dep}=\{D_{dep}^1, \dots, D_{dep}^{\epsilonilonll}\}$ so that each set $U_{dep}^i\subseteqeq V(K_{2n+1})$ is $p_i/2$-random and each set $D_{dep}^i\subseteqeq C(K_{2n+1})$ is $p_i$-random.
Randomly partition $U_{ind}=\{U_{ind}^1, \dots, U_{ind}^{\epsilonilonll}\}, D_{ind}=\{D_{ind}^1, \dots, D_{ind}^{\epsilonilonll}\}$ so that each set $U_{ind}^i$ and $D_{ind}^i$ is $(q/\epsilonilonll)$-random.
Let $\Omega$ be the probability space for $\phi_1$.
We say that $\omega\in \Omega$ is \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and, for each $i\in [\epsilonilonll]$, the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds for $U_{dep}^{i}$, $U_{ind}^{i}$, $D_{dep}^{i}$, $D_{ind}^{i}$, $p=p_i$, $\gamma = q/\epsilonilonll$ and $n=n$.
As $\phi_1$ is a randomized embedding of $T_1$, and by Lemma~\ref{Lemma_sat_matching_random_embedding}, we have that, with high probability, $\omega$ is successful. Note that, for this, we take a union bound over $\epsilonilonll$ events, using that the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds
with probability $1-o(n^{-1})$ in each application.
For each successful $\omega$, define a sequence of trees $T_0^{\omega},T_1^{\omega}, \dots, T_{\epsilonilonll}^{\omega}$. Fix $T_0^{\omega}=T_{\phi_1}(\omega)$ and recursively construct $T_{i+1}^{\omega}$ from $T_i^{\omega}$ by adding an appropriate $(D_{dep}^i\cup D_{ind}^i)$-rainbow size $p_in$ matching into $U_{dep}^i\cup U_{ind}^i$ to make $T_i^\omega$ into a copy of $T_{i+1}$ (which exists as the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds because $\omega$ is successful).
Letting $T_{\phi_{\epsilonilonll}}(\omega)= T_{\epsilonilonll}^{\omega}$ gives the required randomized embedding of $T_{\epsilonilonll}$.
\epsilonilonnd{proof}
The following lemma gives a randomized embedding of any small tree.
\begin{lemma} [Small trees with a replete subset]\label{Lemma_embedding_small_tree}
Let $q\gg \gamma\geq \nu \gg \xi \gg n^{-1}$.
Let $T$ be a tree with $|T|= \gamma n$ containing a set $U\subseteq V(T)$ with $|U|= \nu n$. Let $K_{2n+1}$ be $2$-factorized with $V_\phi, C_\phi$ $q$-random sets of vertices and colours respectively.
Then, there is a randomized rainbow embedding $\phi=(T_\phi, V_\phi, C_\phi)$ along with independent $q/2$-random sets $V_0\subseteqeq V_{\phi}\setminus V(T_{\phi})$, $C_0\subseteqeq C_{\phi}\setminus C(T_{\phi})$ with $(V_0, U_\phi)$ $\xi n$-replete with high probability (where $U_\phi$ is the copy of $U$ in $T_\phi$).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ such that $q\gg \alpha\gg \gamma, \nu$.
Inside $V_{\phi}$ choose disjoint sets $V_0, V_1, V_2$ which are $q/2$-random, $\alpha$-random, and $100\nu/q$-random respectively.
Inside $C_{\phi}$ choose disjoint sets $C_0, C_1, C_2$ which are $q/2$-random, $q/4$-random, and $q/4$-random respectively. Notice that the total number of edges of any particular color is $2n+1$
and the probability that any such edge is going from $V_2$ to $V_1 \cup V_2$ is
$2(100\nu/q)(100\nu/q+\alpha)$. Therefore by Lemmas~\ref{Lemma_high_degree_into_random_set} and \ref{Lemma_Azuma} the following hold with high probability.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item \label{eq_repletetree_1} Every vertex $v$ has $|N_{C_1}(v)\cap V_1|\geq \alpha q n/4\geq 3|T|$ and $|N_{C_2}(v)\cap V_2|\geq (100\nu/q)\cdot qn/4\geq 4|U|$.
\item \label{eq_repletetree_3} Every colour has at most $400\alpha (\nu/q) n\leq 0.1|U|$ edges from $V_2$ to $V_1\cup V_2$.
\epsilonilonnd{enumerate}
Using \ref{eq_repletetree_1}, we can greedily find a rainbow copy $T_\phi$ of $T$ in $V_1\cup V_2\cup C_1\cup C_2$ with $U$ embedded into $V_2$. Indeed, embed one vertex at a time, always embedding a vertex which has one neighbour into preceding vertices. Moreover, put a vertex from $T\setminus U$ into $V_1$ using an edge of colour from $C_1$ and put a vertex from $U$ into $V_2$ using an edge of colour from $C_2$.
Let $U_\phi$ be the copy of $U$ in $T_\phi$. By~\ref{eq_repletetree_3}, and as $|U_\phi|=\nu n$, $(U_\phi, V(K_{2n+1})\setminus (V_1\cup V_2))$ is $0.9\nu n$-replete. Indeed, for every color there are at least $|U_\phi|$ edges of this color incident to $U_\phi$ and at most $0.1|U_\phi|$ of them going to $V_1 \cup V_2$.
Consider a set $V_0'\subseteqeq V(K_{2n+1})$ which is $\big(\frac{1}{1-\alpha-100\nu/q}\big)\cdot (q/2)$-random and independent of $V_1, V_2, C_0, C_1, C_2$. Notice that $V_0'\setminus (V_1\cup V_2)$ is $q/2$-random. Thus the joint distributions of the families of random sets $\{V_0'\setminus (V_1\cup V_2), V_1, V_2, C_0, C_1, C_2\}$ and $\{V_0, V_1, V_2, C_0, C_1, C_2\}$ are exactly the same. By Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random} applied with $A=U_{\phi}, B=V(K_{2n+1})\setminus (V_1\cup V_2), V=V_0'$, the pair $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ is $\xi$-replete with high probability. Since $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ has the same distribution as $(U_{\phi}, V_0)$, the latter pair is also $\xi$-replete with high probability.
\epsilonilonnd{proof}
The following is the main result of this section. It finds a rainbow embedding of every nearly-spanning tree. Given the preceding lemmas, the proof is quite simple. First, we decompose the tree into star forests, matchings of leaves, and connecting paths using Lemma~\ref{Lemma_tree_splitting}. Then, we embed each of these parts using the preceding lemmas in this section.
\begin{proof}[Proof of Theorem~\ref{nearembedagain}]
Choose $\beta$ and $d$ such that $1\geq \epsilonilonps\gg \epsilonilonta\gg \beta \gg \mu\gg d^{-1}\gg \xi$ with $k^{-1}\gg d^{-1}$.
By Lemma~\ref{Lemma_tree_splitting}, there are forests $T_1^{\mathrm{small}}\subseteqeq T_2^{\mathrm{stars}}\subseteqeq T_3^{\mathrm{match}}\subseteqeq T_4^{\mathrm{paths}}\subseteqeq T_5^{\mathrm{match}}=T'$ satisfying \ref{rush1} -- \ref{rush5} with $d=d, n=(1-\epsilonilonps)n$ and $U=U$. Fix $U'=U\cap T_1^{\mathrm{small}}$.
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Consider a set $V_0'\subseteqeq V(K_{2n+1})$ which is $\big(\frac{1}{1-\alpha-100\nu/q}\big)\cdot (q/2)$-random and independent of $V_1, V_2, C_0, C_1, C_2$. Notice that $V_0'\setminus (V_1\cup V_2)$ is $q/2$-random. Thus the joint distributions of the families of random sets $\{V_0'\setminus (V_1\cup V_2), V_1, V_2, C_0, C_1, C_2\}$ and $\{V_0, V_1, V_2, C_0, C_1, C_2\}$ are exactly the same. By Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random} applied with $A=U_{\phi}, B=V(K_{2n+1})\setminus (V_1\cup V_2), V=V_0'$, the pair $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ is $\xi$-replete with high probability. Since $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ has the same distribution as $(U_{\phi}, V_0)$, the latter pair is also $\xi$-replete with high probability.
\epsilonilonnd{proof}
The following is the main result of this section. It finds a rainbow embedding of every nearly-spanning tree. Given the preceding lemmas, the proof is quite simple. First, we decompose the tree into star forests, matchings of leaves, and connecting paths using Lemma~\ref{Lemma_tree_splitting}. Then, we embed each of these parts using the preceding lemmas in this section.
\begin{proof}[Proof of Theorem~\ref{nearembedagain}]
Choose $\beta$ and $d$ such that $1\geq \epsilonilonps\gg \epsilonilonta\gg \beta \gg \mu\gg d^{-1}\gg \xi$ with $k^{-1}\gg d^{-1}$.
By Lemma~\ref{Lemma_tree_splitting}, there are forests $T_1^{\mathrm{small}}\subseteqeq T_2^{\mathrm{stars}}\subseteqeq T_3^{\mathrm{match}}\subseteqeq T_4^{\mathrm{paths}}\subseteqeq T_5^{\mathrm{match}}=T'$ satisfying \ref{rush1} -- \ref{rush5} with $d=d, n=(1-\epsilonilonps)n$ and $U=U$. Fix $U'=U\cap T_1^{\mathrm{small}}$.
Fix $p'=1-|T_2^{\mathrm{stars}}|/n=(|T'|-|T_2^{\mathrm{stars}}|)/n+\epsilonilonps$. We claim that $p'\geq (p+\epsilonilonps)/2$. To see this, let $T''$ be the tree with $|T''|=pn$ formed by deleting leaves of vertices with $\geq k$ leaves in $T'$. Let $A\subseteq V(T_1^{\mathrm{small}})$ be the vertices to which leaves need added to get $T_2^{\mathrm{stars}}$ from $T_1^{\mathrm{small}}$, and, for each $v\in A$, let $d_v\geq d$ be the number of such leaves that are added to $v$. Then we have that $\sum_{v\in A} d_v=|T_2^{\mathrm{stars}}|-|T_1^{\mathrm{small}}|$.
Let $d'_v$ be the number of these leaves added to $v$ which are not themselves leaves in $T'$.
Note that, for each $v\in A$, at least $d_v'$ neighbours of $v$ lie in $V(T')\setminus V(T_2^{\mathrm{stars}})$. Thus, we have
$\sum_{v\in A}d_v'\leq |T'|-|T_2^{\mathrm{stars}}|$. Moreover, for each $v\in A$ with $d_v-d'_v \geq k$ we deleted $d_v-d'_v$ leaves attached to $v$ when we formed $T''$ from $T'$, otherwise we deleted nothing. In both cases we deleted at least $d_v-d'_v-k$ leaves for each $v \in A$.
Therefore, using that $|A| \leq|T_1^{\mathrm{small}}|\leq n/d$, we have
$$|T'|-|T''| \geq \sum_{v\in A}(d_v-d_v'-k)=|T_2^{\mathrm{stars}}|-|T_1^{\mathrm{small}}|-\sum_{v\in A}(d_v'+k) \geq |T_2^{\mathrm{stars}}|-(k+1)n/d-\sum_{v\in A}d_v'\,.$$
Thus,
$$
pn= |T''| \leq |T'|-|T_2^{\mathrm{stars}}|+(k+1)n/d+\sum_{v\in A}d_v'\leq 2(|T'|-|T_2^{\mathrm{stars}}|)+\epsilonilonps n= 2p'n-\epsilonilonps n,
$$
implying, $p'\geq (p+\epsilonilonps)/2$.
For each $i\in \{2, \dots, 5\}$ (and the label $*$ meaning ``$\mathrm{small},\, \mathrm{stars},\, \mathrm{match}$ or $\mathrm{paths}$" appropriately), let $p_i=(|T_i^*|-|T_{i-1}^*|)/n$. Set $p_6=\epsilonilonpsilon-\beta.$
These constants represent the proportion of colours which are used for embedding each of the subtrees (with $p_6$ representing the proportion of colours left over for the set $C$ in the statement of the lemma).
Notice that $p'=p_3+p_4+p_5+p_6+\beta$.
For each $i\in [6]$, choose disjoint, independent $2\mu$-random sets $V^{i}_{ind}, C_{ind}^i$. Do the following.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_embedding_small_tree} to $T_1^{\mathrm{small}}$, $U=U'$, $V_\phi=V_{1}^{ind}$, $C_\phi=C_{1}^{ind}$, $q=2\mu$, $\gamma=|T_1^{small}|/n$, $\nu=|U'|/n$ and $\xi=\xi$.
This gives a randomized rainbow embedding $\phi_1=(T_{\phi_1}, V_{1}^{ind}, C_{1}^{ind})$ of $T_1^{small}$ and $\mu$-random, independent sets $V_0\subseteqeq V_{1}^{ind}\setminus V(T_{\phi_1})$, $C_0\subseteqeq C_{1}^{ind}\setminus C(T_{\phi_1})$ with $(U'_{\phi_1},V_0)$ $\xi n$-replete with high probability (where $U'_{\phi_1}$ is the copy of $U'$ in $T_{\phi_1}$).
\epsilonilonnd{itemize}
Notice that $\phi_1'=(V_1^{ind}\cup \dots \cup V_6^{ind}, C_1^{ind}\cup \dots\cup C_6^{ind}, T_{\phi_1} )$ is also a randomized embedding of $T_1^{\mathrm{small}}$.
Then, do the following.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_extending_with_large_stars} with $T_1=T_1^{\mathrm{small}}$, $T_2=T_2^{\mathrm{stars}}$, $\phi_1=\phi_1'$, $p=p', \beta=\beta, \gamma=12\mu, d=d, n=n$. This gives a randomized embedding $\phi_{2}=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2^{\mathrm{stars}}$ extending $\phi_{1}'$, with $V_{dep}=V(K_{2n+1})\setminus V_{\phi_2}$, $C_{dep}=C(K_{2n+1})\setminus C_{\phi_2}$ $(p'-\beta)$-random sets of vertices/colours.
\epsilonilonnd{itemize}
Notice that $\phi_2'=(V_{\phi_2}\setminus (V_3^{ind}\cup \dots \cup V_6^{ind}), C_{\phi_2}\setminus (C_3^{ind}\cup \dots \cup C_6^{ind}), T_{\phi_2})$ is also a randomized embedding of $T_2^{\mathrm{stars}}$. Indeed, recall that by our definition of extension we embed vertices of $T_2^{\mathrm{stars}}\setminus T_1^{small}$ outside of the set $V_1^{ind}\cup \dots \cup V_6^{ind}$ using the edges whose colors are also outside of $C_1^{ind}\cup \dots\cup C_6^{ind}$. Moreover, by our constriction, $V(T_{\phi_1})$ is disjoint from
$V_2^{ind}\cup \dots \cup V_6^{ind}$ and $C(T_{\phi_1})$ is disjoint from
$C_2^{ind}\cup \dots\cup C_6^{ind}$.
Using that $p_3+p_4+p_5+p_6 = p'-\beta$ and Lemma~\ref{Lemma_mixture_of_p_random_sets}, randomly partition $C_{dep}= C_{dep}^3\cup C_{dep}^4\cup C_{dep}^5\cup C_{dep}^6$ with the sets $C_{dep}^i$ $p_i$-random subsets of $C(K_{2n+1})$.
Using that $p'-\beta= (p_3+p_4+p_5+p_6)/2 + (p'-\beta)/2\geq p_3/2+p_4/2+p_5/2+ (p+\epsilonilonps)/6$, inside $V_{dep}$ we can choose disjoint sets $V_{dep}^3, V_{dep}^4, V_{dep}^5, V_{dep}^6$ which are $\rho$-random for $\rho=p_3/2, p_4/2, p_5/2, (p+\epsilonilonps)/6$ respectively. We remark that some of the random sets ($V_{ind}^2, C_{ind}^2, V_{ind}^6, C_{ind}^6, V_{dep}^4,$ and $C_{dep}^4$) will not actually used in the embedding. They are only there because it is notationally convenient to allocate equal amounts of vertices/colours for the different parts of the embedding.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_extending_with_matchings} with $T_{\epsilonilonll}=T_3^{\mathrm{match}}, T_1=T_{2}^{\mathrm{stars}}$, $V_{dep}=V_{dep}^3$, $C_{dep}=C_{dep}^3$, $V_{ind}=V_{ind}^3$, $C_{ind}=C_{ind}^3$, $\phi_1=\phi_2'$, $\epsilonilonll=d, p=p_3, q=2\mu, n=n$. This gives we get a randomized embedding $\phi_3=(V_{\phi_3}, C_{\phi_3}, T_{\phi_3})$ of $T_3^{\mathrm{match}}$ into $V^3_{ind}\cup V^3_{dep}\cup V_{\phi_2'}$, $V^3_{dep}\cup V^3_{ind}\cup C_{\phi_2'}$ extending $\phi_2$.
\item Apply Lemma~\ref{Lemma_extending_with_connecting_paths} with $T_2=T_4^{\mathrm{paths}}, T_1=T_{3}^{\mathrm{match}}$, $V_{ind}=V_{ind}^4$, $C_{ind}=C_{ind}^4$, $\phi_1=\phi_3$, $p=2\mu, q=(2k)^{-1}, n=n$. This gives a randomized embedding $\phi_4=(V_{\phi_4}, C_{\phi_4}, T_{\phi_4})$ of $T_4^{\mathrm{paths}}$ into $V^4_{ind}\cup V_{\phi_3}, C^4_{ind}\cup C_{\phi_3}$ extending $\phi_3$.
\item By Lemma~\ref{Lemma_extending_with_matchings} with $T_{\epsilonilonll}=T_5^{\mathrm{match}}, T_1=T_{4}^{\mathrm{paths}}$, $V_{dep}=V_{dep}^5$, $C_{dep}=C_{dep}^5$, $V_{ind}=V_{ind}^5$, $C_{ind}=C_{ind}^5$, $\phi_1=\phi_4$, $\epsilonilonll=d, p=p_5, q=2\mu, n=n$, we get a randomized embedding $\phi_5=(V_{\phi_5}, C_{\phi_5}, T_{\phi_5})$ of $T_5^{\mathrm{match}}$ into $V^5_{ind}\cup V^5_{dep}\cup V_{\phi_4}$, $C^5_{dep}\cup C^5_{ind}\cup V_{\phi_4}$ extending $\phi_4$.
\epsilonilonnd{itemize}
Now the lemma holds with $\hat T'=T_{\phi_5}$, $V_0=V_0$, $C_0=C_0$ (recall that these sets are from the application of Lemma~\ref{Lemma_embedding_small_tree} above), $V=V_{dep}^6$, and $C\subseteqeq C_{dep}^6$ a $(1-\epsilonilonta)\epsilonilonps$-random subset.
\epsilonilonnd{proof}
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\section{The embedding in Case C}\label{sec:lastC}\label{sec:caseC}
For our embedding in Case C, we distinguish between those trees with one very high degree vertex, and those trees without. The first case is covered by Theorem~\ref{Theorem_one_large_vertex}. In this section, we prove Lemma~\ref{Lemma_small_tree}, which covers the second case and thus completes the proof of Theorem~\ref{Theorem_case_C}.
For trees with no very high degree vertex, we start with the following lemma which embeds a small tree in the $ND$-colouring in a controlled fashion. In particular, we wish to embed a prescribed small portion of the tree into an interval in the cyclic ordering of $K_{2n+1}$ so that the distance between any two consecutive vertices in the image of the embedding is small.
\begin{lemma}[Embedding a small tree into prescribed intervals]\label{Lemma_small_tree}
Let $n\geq 10^5$.
Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be a tree with at most $n/100$ vertices whose vertices are partitioned as $V(T)=V_0\cup V_1\cup V_2$ with $|V_1|, |V_2|\leq 2n/\log^4 n$.
Let $I_0, I_1, I_2$ be disjoint intervals in $V(K_{2n+1})$ with $|I_0|\geq 7|V_0|$, $|I_1|\geq 8|V_1|\log^3 n $ and $|I_2|\geq 8|V_2|\log^3 n $.
Then, there is a rainbow copy $S$ of $T$ in $K_{2n+1}$ such that, for each $0\leq i\leq 2$, the image of $V_i$ is contained in $I_i$, and, furthermore, any consecutive pair of vertices of $S$ in $I_1$, or in $I_2$, are within distance $16\log^3 n$ of each other.
\epsilonilonnd{lemma}
\begin{proof}
Let $k\in \{\lfloor 2\log n\rfloor,\lfloor 2\log n\rfloor-1\}$ be odd.
Choose consecutive subintervals $I_1^{1}, \dots, I_1^{|V_1|}$ in $I_1$ of length $k^3$, and choose consecutive subintervals $I_2^{1}, \dots, I_2^{|V_2|}$ in $I_2$ of length $k^3$.
Partition $C(K_{2n+1})$ as $C_0\cup C_1^{1}\cup \dots\cup C_1^{k}\cup C_2^{1}\cup \dots\cup C_2^{k}$ so that, for each $m\in [n]$ and $i\in [k]$,
\[
m\in \left\{\begin{array}{ll}
C_0 & \text{ if }m\text{ is odd},\\
C_1^i & \text{ if }m\text{ is even and }m\epsilonilonquiv i\mod{2k+1},\\
C_2^i & \text{ if }m\text{ is even and }m\epsilonilonquiv i+k\mod{2k+1}.
\epsilonilonnd{array}
\right.
\]
Note that, for every $p\in [2]$, $i\in [k]$, $j\in [|V_p|]$, and $v\notin I_p^j$, since $C_p^i$ contains only even colours and $k$ is odd, $v$ has at least $\lfloor |I_p^j|/2(2k+1)\rfloor\geq k$ colour-$C_p^i$ neighbours in $I_p^j$ (we call a vertex $u$ a colour-$C_p^i$ neighbour of $v$ if the colour of the edge $vu$ belongs to $C_p^i$).
Order $V(T)$ arbitrarily as $v_1, \dots, v_{m}$ such that $\{v_1, \dots, v_i\}$ forms a subtree of $T$ for each $i\in [m]$.
We embed the vertices $v_1, \dots, v_{m}$ one by one, in $m$ steps, so that, in step $i$, we embed $v_i$ to some $u_i\in V(K_{2n+1})$. Since $T[v_1, \dots, v_i]$ is a tree, for each $i$, there is a unique $f(i)<i$ for which $v_{f(i)}v_i\in E(T)$. We will maintain that the edges $u_iu_{f(i)}$ have different colours for each $i$.
At step $i$ (the step in which $u_1, \dots, u_{i-1}$ have already been chosen and we are choosing $u_i$) we say that a vertex $x$ is \epsilonilonmph{free} if $x\not\in\{u_1, \dots, u_{i-1}\}$. We say that an interval $I_s^t$ is \epsilonilonmph{free}, if it contains no vertices from $\{u_1, \dots, u_{i-1}\}$. We say that a colour is \epsilonilonmph{free} if it does not occur on $\{u_{f(j)}u_j:2\leq j\leq i-1\}$. The procedure to choose $u_i$ at step $i$ is as follows.
\begin{itemize}
\item If $v_i\in V_0$, then $u_i$ is embedded to any free vertex in $I_0$ so that, if $i\geq 2$, $u_{f(i)}u_i$ is any free colour in $C_0$.
\item For $p\in [2]$, if $v_i\in V_p$, then $u_i$ is embedded into any free interval $I_p^i$ so that, if $i\geq 2$, $u_{f(i)}u_i$ uses any free colour in $C_p^j$, for $j$ as small as possible.
\epsilonilonnd{itemize}
The lemma follows if we can embed all the vertices in this way. Indeed, as we have exactly the right number of intervals to embed one vertex per interval, consecutive vertices in $I_1$ (or $I_2$) are in consecutive intervals $I_1^i$, and hence at most $16\log^3n$ apart. Since $C_0$ contains all the odd colours,
vertex $u_{f(i)}$ has $\geq \lfloor |I_0|/2\rfloor-1 > 3|V_0|$ colour-$C_0$ neighbours in $I_0$. At each step at most $|V_0|$ of the vertices in $I_0$ are occupied, and at most $|V_0|$ colours are used. Since the $ND$-colouring is locally $2$-bounded,
this forbids at most $3|V_0|$ vertices. Therefore, there is always room to embed each $v_i \in V_0$ into $I_0$.
To finish the proof of the lemma, it is sufficient to show that, throughout the process, for each $p\in [2]$, there will always be sets $C_p^j$ which are never used.
\begin{claim}
Let $p\in [2]$, $i\in [m]$ and $s\in [k]$.
At step $i$, if there are $\geq |V_p|/2^{s}$ free intervals $I_p^j$, then no colour from $C_p^s$ has been used up to step $i$.
\epsilonilonnd{claim}
\begin{proof}
The proof is by induction on $i$. The initial case $i=0$ is trivial.
Suppose it is true for steps $1,\ldots,i-1$ and suppose that there are still at least $|V_p|/2^s$ free intervals $I_p^j$. Now, by the induction hypothesis for $p'=p$, $i'=i-1$ and $s'=s-1$, colours from $C_p^{s-1}$ were only used when there were fewer than $|V_p|/2^{s-1}$ free intervals $I_p^j$, and hence at most $|V_p|/2^{s-1}$ vertices were embedded using colours from $C_p^{s-1}$. In each free interval, $u_{f(i)}$ has at least $k$ colour-$C_p^{s-1}$ neighbours.
and hence at least $k|V_p|/2^s$ colour-$C_p^{s-1}$ neighbours in the free intervals in total. As the $ND$-colouring is locally 2-bounded, this gives at least
$k|V_p|/2^{s+1}> |V_p|/2^{s-1}$ colour-$C_p^{s-1}$ neighbours adjacent to $u_{f(i)}$ by edges of different colours. Therefore $u_{f(i)}$ has some colour $C_p^{s-1}$-neighbour $u_i$ in a free interval so that $u_{f(i)}u_i$ is a free colour. Thus, no colour from $C_p^s$ is used in step $i$.
\epsilonilonnd{proof}
Taking $s=k$, we see that colours from $C_1^k$ and $C_2^k$ never get used while there is at least one free interval. As there are exactly as many intervals as vertices we seek to embed, the colours from $C_1^k$ and $C_2^k$ are never used, and hence the procedure successfully embeds $T$.
\epsilonilonnd{proof}
\begin{figure}[h]
\begin{center}
{
\begin{tikzpicture}[scale=0.7,define rgb/.code={\definecolor{mycolor}{rgb}{#1}},
rgb color/.style={define rgb={#1},mycolor}]]
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\begin{figure}[h]
\begin{center}
{
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rgb color/.style={define rgb={#1},mycolor}]]
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\draw [black] (4.9315068493*-9:5.7) node {\small $I_2$};
\draw [rotate=4.9315068493*-45,black] (0:5.1) -- (0:5.6);
\draw [rotate=4.9315068493*-45,black] (0:{6.25-0.1}) node {\small $1.91n$};
\draw [rotate=4.9315068493*-12,black] (0:5.1) -- (0:5.6);
\draw [rotate=4.9315068493*-12,black] (0:6.25) node {\small $0.96n$};
\draw [rotate=4.9315068493*-45,black] (5.35,0) arc (0:4.9315068493*33:5.35);
\epsilonilonnd{tikzpicture}
}
\epsilonilonnd{center}
\caption{Embedding the tree in Case C.}\label{Figure_many_vertex}
\epsilonilonnd{figure}
Now we are ready to prove Case C. Letting $T'$ be a tree $T$ in Case C with the neighbouring leaves of vertices with $\geq \log^4 n$ leaves removed, we embed $T'$ using the above lemma.
Then, much like in the proof Theorem~\ref{Theorem_one_large_vertex}, leaves are added to high degree vertices one at a time. In order to make sure we use all the colours, these leaves are located in particular intervals of the ND-colouring. In the proof, for technical reasons, we have three different kinds of high degree vertices (called type 1, type 2 and type 3 vertices), with vertices of a particular type having its neighbours chosen by a particular rule. (We note that we may have no type 2 vertices.) We embed the vertices of different types into 3 disjoint intervals.
\begin{proof}[Proof of Theorem~\ref{Theorem_case_C}]
See Figure~\ref{Figure_many_vertex} for an illustration of this proof. If $d_i\geq 2n/3$ for some $i$, then the result follows from Theorem~\ref{Theorem_one_large_vertex}.
Assume then that $v_1, \dots, v_{\epsilonilonll}$ are ordered so that $2n/3\geq d_1\geq \dots\geq d_{\epsilonilonll}$. Note that $d_1+\dots+d_{\epsilonilonll}= n+1-|T'|\geq 99n/100+1$. Choose the smallest $m$ with $0.05n< d_1+ \dots+ d_m$. From minimality, we have $d_1+\dots+ d_m\leq d_1+0.05n$. Let $V_1=\{v_1, \dots ,v_m\}$, $V_2=\{v_{m+1}, \dots, v_{\epsilonilonll}\}$, and $V_0=V(T')\setminus (V_1\cup V_2)$.
Let $I_0=[0.83n, 0.9n]$, $I_1=[0.7n, 0.71n]$, and $I_2=[0.91n, 0.92n]$. Then, it is easy to check that $|I_0| \geq 7|V_0|$, $|I_1|\geq 8|V_1|\log^3 n$ and $|I_2|\geq 8|V_2|\log^3 n$.
By Lemma~\ref{Lemma_small_tree}, there is a rainbow embedding of $T'$ in $I_0\cup I_1\cup I_2$ with $V_i$ embedded to $I_i$, for each $0 \leq i\leq 2$, and consecutive embedded vertices in $I_1$, and in $I_2$, within distance $16\log^3 n$ of each other. Say the copy of $T'$ from this is $S'$.
For odd $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m},u_{m-1},u_{m-3},\ldots,u_2.
\]
For even $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m-1},u_{m},u_{m-2},\ldots,u_2.
\]
Relabel the vertices in $V_2$ so that, if $u_i$ is the image of $v_i$ under the embedding, then for each $i\in [\epsilonilonll]\setminus [m]$, the vertices in the image of $V_2$ appear in $I_2$ in the order $u_{m+1},\ldots,u_\epsilonilonll$. Finally, relabel the integers $d_i$, $i\in [\epsilonilonll]$, to match this relabelling of the vertices $v_i$, $i\in [\epsilonilonll]$. We uses this new labelling to embed neighbours of $u_i, i \leq \epsilonilonll$ into three disjoint intervals (that are also disjoint from $V(S')$): $[1,u_1)$ if $i\leq m$ is odd, $(u_2,0.82n]$ if $i\leq m$ is even and $[0.96n,1.92n]$ if $m < i \leq \epsilonilonll$.
For each $1\leq i\leq m$, if $i$ is odd we say that $u_i$ is \epsilonilonmph{type 1}, and if $i$ is even we say that $u_i$ is \epsilonilonmph{type 2}. For each $m<i\leq \epsilonilonll$, we say that $u_i$ is \epsilonilonmph{type 3}. Note that, if $m=1$, then there are no type 2 vertices.
Furthermore, note that, if $m=1$, then $V(S')\subseteq \{u_1\}\cup I_0\cup I_2\subseteq \{u_1\}\cup [0.83n,0.92n]$, while, if $m>1$, then $V(S')\subseteq [u_1,u_2]\cup I_0\cup I_2\subseteq [u_1,u_2]\cup [0.83n,0.92n]$.
Let $C(K_{2n+1})\setminus C(S')=C_1\cup \ldots \cup C_\epsilonilonll$ so that, for each $i\in [\epsilonilonll]$, $|C_i|=d_i$, and, if $i<j$, then all the colours in $C_i$ are smaller than all the colours in $C_j$. Note that this is possible as $|C(K_{2n+1})\setminus C(S')|=n-|E(T')|=d_1+\ldots+d_\epsilonilonll$.
Futhermore, note that, if $i,j\in [\epsilonilonll]$ with $i\leq j-2$, then $\max(C_i)\leq \min (C_j)-d_{i+1}\leq \min(C_j)-\log^4 n$.
For each $i\in [\epsilonilonll]$, do the following.
\begin{itemize}
\item If $u_i$ is a type 1 vertex, embed the vertices in $U_i:=\{u_i-c:c\in C_{i}\}$ as neighbours of $u_i$.
\item If $u_i$ is a type 2 or 3 vertex, embed the vertices in $U_i:=\{u_i+c:c\in C_{i}\}$ as neighbours of $u_i$.
\epsilonilonnd{itemize}
Notice that, from the definition of the ND-colouring, we use edges with colour in $C_i$ to attach the neighbours of $u_i$, for each $i\in [\epsilonilonll]$. Therefore, by design, the graph formed is rainbow. We need to show that the sets $U_i$, $i\in [\epsilonilonll]$, are disjoint from each each other and from $V(S')$, so that we have a copy of $T$.
\begin{claim} If $m=1$, then $\max(C_m)\leq 0.68n$, while, if $m>1$, then $\max(C_m)\leq 0.11n$.
\epsilonilonnd{claim}
\begin{proof}
If $m=1$, then, as at most $0.01n$ colours are used to embed $T'$ and $d_1\leq 2n/3$, we have $\max(C_m)\leq 0.67n+0.01n=0.68n$.
Suppose then that $m>1$, and hence $d_1<0.05n$. By the minimality in the choice of $m$, we have that $d_1+\ldots+d_m\leq 0.05n+0.05n=0.1n$. Therefore, $\max(C_m)\leq 0.1n+0.01n= 0.11n$.
\epsilonilonnd{proof}
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By Lemma~\ref{Lemma_small_tree}, there is a rainbow embedding of $T'$ in $I_0\cup I_1\cup I_2$ with $V_i$ embedded to $I_i$, for each $0 \leq i\leq 2$, and consecutive embedded vertices in $I_1$, and in $I_2$, within distance $16\log^3 n$ of each other. Say the copy of $T'$ from this is $S'$.
For odd $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m},u_{m-1},u_{m-3},\ldots,u_2.
\]
For even $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m-1},u_{m},u_{m-2},\ldots,u_2.
\]
Relabel the vertices in $V_2$ so that, if $u_i$ is the image of $v_i$ under the embedding, then for each $i\in [\epsilonilonll]\setminus [m]$, the vertices in the image of $V_2$ appear in $I_2$ in the order $u_{m+1},\ldots,u_\epsilonilonll$. Finally, relabel the integers $d_i$, $i\in [\epsilonilonll]$, to match this relabelling of the vertices $v_i$, $i\in [\epsilonilonll]$. We uses this new labelling to embed neighbours of $u_i, i \leq \epsilonilonll$ into three disjoint intervals (that are also disjoint from $V(S')$): $[1,u_1)$ if $i\leq m$ is odd, $(u_2,0.82n]$ if $i\leq m$ is even and $[0.96n,1.92n]$ if $m < i \leq \epsilonilonll$.
For each $1\leq i\leq m$, if $i$ is odd we say that $u_i$ is \epsilonilonmph{type 1}, and if $i$ is even we say that $u_i$ is \epsilonilonmph{type 2}. For each $m<i\leq \epsilonilonll$, we say that $u_i$ is \epsilonilonmph{type 3}. Note that, if $m=1$, then there are no type 2 vertices.
Furthermore, note that, if $m=1$, then $V(S')\subseteq \{u_1\}\cup I_0\cup I_2\subseteq \{u_1\}\cup [0.83n,0.92n]$, while, if $m>1$, then $V(S')\subseteq [u_1,u_2]\cup I_0\cup I_2\subseteq [u_1,u_2]\cup [0.83n,0.92n]$.
Let $C(K_{2n+1})\setminus C(S')=C_1\cup \ldots \cup C_\epsilonilonll$ so that, for each $i\in [\epsilonilonll]$, $|C_i|=d_i$, and, if $i<j$, then all the colours in $C_i$ are smaller than all the colours in $C_j$. Note that this is possible as $|C(K_{2n+1})\setminus C(S')|=n-|E(T')|=d_1+\ldots+d_\epsilonilonll$.
Futhermore, note that, if $i,j\in [\epsilonilonll]$ with $i\leq j-2$, then $\max(C_i)\leq \min (C_j)-d_{i+1}\leq \min(C_j)-\log^4 n$.
For each $i\in [\epsilonilonll]$, do the following.
\begin{itemize}
\item If $u_i$ is a type 1 vertex, embed the vertices in $U_i:=\{u_i-c:c\in C_{i}\}$ as neighbours of $u_i$.
\item If $u_i$ is a type 2 or 3 vertex, embed the vertices in $U_i:=\{u_i+c:c\in C_{i}\}$ as neighbours of $u_i$.
\epsilonilonnd{itemize}
Notice that, from the definition of the ND-colouring, we use edges with colour in $C_i$ to attach the neighbours of $u_i$, for each $i\in [\epsilonilonll]$. Therefore, by design, the graph formed is rainbow. We need to show that the sets $U_i$, $i\in [\epsilonilonll]$, are disjoint from each each other and from $V(S')$, so that we have a copy of $T$.
\begin{claim} If $m=1$, then $\max(C_m)\leq 0.68n$, while, if $m>1$, then $\max(C_m)\leq 0.11n$.
\epsilonilonnd{claim}
\begin{proof}
If $m=1$, then, as at most $0.01n$ colours are used to embed $T'$ and $d_1\leq 2n/3$, we have $\max(C_m)\leq 0.67n+0.01n=0.68n$.
Suppose then that $m>1$, and hence $d_1<0.05n$. By the minimality in the choice of $m$, we have that $d_1+\ldots+d_m\leq 0.05n+0.05n=0.1n$. Therefore, $\max(C_m)\leq 0.1n+0.01n= 0.11n$.
\epsilonilonnd{proof}
From the claim, as the vertices $u_1,\ldots,u_m$ are in $[0.7n,0.71n]$ and $\max(C_i) \leq \max (C_m)$, for each $i\in [m]$, for each type 1 vertex $u_{i}$, we have $U_{i}\subseteq [u_{i}-\max (C_i),u_i-\min(C_i)]\subseteq [0.02n,0.71n] \subseteq [1,0.71n]$. For each type 2 vertex $u_{i}$, we have $U_{i}\subseteq [u_{i}+\min (C_i),u_{i}+\max(C_i)]\subseteq [0.7n,0.82n]$.
Now, for each type 3 vertex $u_{i}$, we have, as $i>m$, that $\min(C_i)\geq 0.05n$. Therefore, as $\max(C_i)\leq n$, we have $U_{i}\subseteq [u_{i}+\min (C_i),u_{i}+\max(C_i)]\subseteq [0.96n,1.92n]$. Furthermore, by the ordering of $u_{m+1},\ldots,u_\epsilonilonll$, for each $m+1\leq i<j\leq \epsilonilonll$, we have $u_i<u_j$ and $\max(C_i)<\min(C_j)$ so that
$u_{i}+\max (C_i)< u_{j}+\min (C_j)$. Thus, the sets $U_i$, $m+1\leq i\leq \epsilonilonll$ are disjoint sets in $[0.96n,1.92n]$.
\begin{claim}
If $u_{i}$ is a type 1 vertex with $i\leq m-2$, then $\min(U_i)< \max (U_{i+2})$.
If $u_{i}$ is a type 2 vertex with $i\leq m-2$, then $\max(U_i) < \min (U_{i+2})$.
\epsilonilonnd{claim}
\begin{proof}
If $u_i$ is a type 1 vertex with $i\leq m-2$, then $u_{i+2}$ is also a type 1 vertex, which appears consecutively with $u_i$ in the interval $I_1$ in the embedding of $T'$, and hence $u_{i+2}-u_i\leq 16 \log^3 n<\log^4 n$. Therefore,
\[
\min(U_i)=u_{i}-\max (C_i)\geq u_i-\min(C_{i+2})+\log^4 n > u_{i+2}-\min(C_{i+2})=\max(U_{i+2}).
\]
On the other hand, if $u_i$ is a type 2 vertex with $i\leq m-2$, then $u_{i+2}$ is also a type 2 vertex, which appears consecutively with $u_i$ in the interval $I_1$ in the embedding of $T'$, and hence $u_{i}-u_{i+2}\leq 16 \log^3 n <\log^4 n$. Therefore,
\[
\max(U_i)=u_{i}+\max (C_i)\leq u_i+\min(C_{i+2})-\log^4 n < u_{i+2}+\min(C_{i+2})=\min(U_{i+2}).\qed
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here
\]
\epsilonilonnd{proof}
By the claim, the sets $U_i$, $i\leq m$ and $i$ is odd, are disjoint and lie in the interval $[1,\max(U_1)]\subseteq [1,u_1)$. If $m\geq 2$, then, by the claim again, the sets $U_i$, $i\leq m$ and $i$ is even, are disjoint and lie in the interval $[\min(U_2),0.82n]\subseteq (u_2,0.82n]$.
In summary, we have shown the following.
\begin{itemize}
\item The sets $U_i$, $i\leq m$ and $i$ is odd are disjoint sets in $[1,u_1)$.
\item If $m\geq 2 $, then the sets $U_i$, $i\leq m$ and $i$ is even, are disjoint sets in $(u_2,0.82n]$.
\item The sets $U_i$, $m<i\leq \epsilonilonll$, are disjoint sets in $[0.96n,1.92n]$.
\item If $m=1$, then $V(S')\subseteq \{u_1\}\cup [0.83n,0.92n]$, while, if $m\geq 2$, then $V(S')\subseteq [u_1,u_2]\cup [0.83n,0.92n]$.
\epsilonilonnd{itemize}
Thus, both when $m=1$ and when $m\geq 2$, we have that $U_i$, $i\in [\epsilonilonll]$, are disjoint sets in $[2n+1]\setminus V(S')$, as required.
\epsilonilonnd{proof}
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\section{Concluding remarks}\label{sec:conc}
Many powerful techniques have been developed to decompose graphs into bounded degree graphs~\cite{bottcher2016approximate, messuti2016packing, ferber2017packing, kim2016blow,joos2016optimal}.
On the other hand, all these techniques encounter some barrier when dealing with trees with arbitrarily large degrees.
Having overcome this ``bounded degree barrier'' for Ringel's Conjecture, we hope that our ideas might be useful for other problems as well. Here we mention two such questions.
The closest relative of Ringel's conjecture is the following conjecture on graceful labellings, which is also mentioned in the introduction. This is a very natural problem to apply our techniques to.
\begin{conjecture}[K\"otzig-Ringel-Rosa, \cite{rosa1966certain}]\label{Conjecture_Graceful}
The vertices of every $n$ vertex tree $T$ can be labelled by the numbers $1, \dots, n$, such that the differences $|u-v|$, $uv\in E(T)$, are distinct.
\epsilonilonnd{conjecture}
\noindent
This conjecture was proved for many isolated classes of trees like caterpillars, trees with $\leq 4$ leaves, firecrackers, diameter $\leq 5$ trees, symmetrical trees, trees with $\leq 35$ vertices, and olive trees (see Chapter 2 of \cite{gallian2009dynamic} and the references therein).
Conjecture~\ref{Conjecture_Graceful} is also known to hold asymptotically for trees of maximum degree at most $n/\log n$ \cite{adamaszek2016almost} but solving it for general trees, even asymptotically, is already wide open.
Another very interesting related problem is the G\'yarf\'as Tree Packing Conjecture. This also concerns decomposing a complete graph into specified trees, but the trees are allowed to be different from each other.
\begin{conjecture}[Gy\'arf\'as, \cite{gyarfas1978packing}]\label{Conjecture_Gyarfas}
Let $T_1, \dots, T_{n}$ be trees with $|T_i|=i$ for each $i\in [n]$. The edges of $K_n$ can be decomposed into $n$ trees which are isomorphic to $T_1, \dots, T_{n}$ respectively.
\epsilonilonnd{conjecture}
\noindent
This conjecture has been proved for bounded degree trees by Joos, Kim, K{\"u}hn and Osthus~\cite{joos2016optimal} but in general it is wide open.
It would be interesting to see if any of our techniques can be used here to make further progress.
\noindent
{\bf Acknowledgements.} Parts of this work were carried out when the first two authors visited the Institute for Mathematical Research (FIM) of ETH Zurich. We would like to thank FIM for its hospitality and for creating a stimulating research environment.
\epsilonilonnd{document}
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\begin{document}
\title[Notes on the codimension one conjecture]{Notes on the codimension one conjecture in the operator corona theorem}
\author{Maria F. Gamal'}
\address{
St. Petersburg Branch\\ V. A. Steklov Institute
of Mathematics\\
Russian Academy of Sciences\\ Fontanka 27, St. Petersburg\\
191023, Russia
}
\email{[email protected]}
\thanks{Partially supported by RFBR grant No. 14-01-00748-a}
\subjclass[2010]{ Primary 30H80; Secondary 47A45, 47B20.}
\keywords{Operator corona theorem, contraction, similarity to an isometry}
\begin{abstract}
Answering on the question of S.R.Treil \cite{23}, for every $\delta$, $0<\delta<1$,
examples of contractions are constructed such that their characteristic functions
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfy the conditions $$\|F(z)x\|\geq\delta\|x\| \ \text{ and } \
\dim\mathcal E_\ast\ominus F(z)\mathcal E =1 \ \text{ for every } \ z\in\mathbb D, \ \ x\in\mathcal E,$$
but $F$ are not
left invertible. Also, it is shown that the condition
$$\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\frak S_1}<\infty,$$
where $\frak S_1$ is the trace class of operators, which is sufficient for the
left invertibility
of the operator-valued function $F$ satisfying the estimate $\|F(z)x\|\geq\delta\|x\|$
for every $z\in\mathbb D$, $x\in\mathcal E$, with some $\delta>0$ (S.R.Treil, \cite{22}), is necessary
for the left invertibility of an inner function $F$ such that $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$
for some $z\in\mathbb D$.
\end{abstract}
\maketitle
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\section{Introduction}
Let $\mathbb D$ be the open unit disk, let $\mathbb T$ be the unit circle,
and let $\mathcal E$ and $\mathcal E_\ast$ be separable Hilbert spaces.
The space $H^\infty(\mathcal E\to\mathcal E_\ast)$ is the space of bounded analytic functions on $\mathbb D$
whose values are (linear, bounded) operators acting from $\mathcal E$ to $\mathcal E_\ast$.
If $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$, then $F$ has nontangential boundary values
$F(\zeta)$ for a.e. $\zeta\in\mathbb T$ with respect to the Lebesgue measure $m$ on $\mathbb T$.
Every function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ has the inner-outer
factorization, that is, there exist an auxiliary Hilbert space $\mathcal D$ and two functions
$\Theta\in H^\infty(\mathcal E\to\mathcal D)$ and $\Omega\in H^\infty(\mathcal D\to\mathcal E_\ast)$ such that
$F=\Theta\Omega$, $\Theta$ is inner, that is, $\Theta(\zeta)$ is an isometry for a.e. $\zeta\in\mathbb T$,
and $\Omega$ is outer. The definition of outer function is not recalled here,
but it need to mentioned that
for an outer function $\Omega$, $\operatorname{clos}\Omega(z)\mathcal D=\mathcal E_\ast$
for all $z\in\mathbb D$ and for a.e. $z\in\mathbb T$. Recall that a function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is called $\ast$-inner
($\ast$-outer), if the function $F_\ast\in H^\infty(\mathcal E_\ast\to\mathcal E)$,
$F_\ast(z)=F^\ast(\overline z)$, $z\in\mathbb D$, is inner (outer) (see {\cite[Ch.V]{16}}, also {\cite[\S A.3.11.5]{14}}).
Every analytic operator-valued function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ such that $\|F\|\leq 1$ can be represented as an orthogonal sum of a unitary constant and a purely contractive function $F_0$, that is,
there exist representations of Hilbert spaces $\mathcal E=\mathcal E'\oplus\mathcal E_0$ and
$\mathcal E_\ast=\mathcal E_\ast'\oplus\mathcal E_{\ast 0}$ and a unitary operator
$W\colon\mathcal E'\to\mathcal E_\ast'$ such that $F(z)\mathcal E'\subset\mathcal E_\ast'$,
$F(z)|_{\mathcal E'}=W$, $F(z)\mathcal E_0\subset\mathcal E_{\ast 0}$, $F_0(z)=F(z)|_{\mathcal E_0}$ for every
$z\in\mathbb D$, and $\|F_0(0)x\|<\|x\|$ for every $x\in\mathcal E_0$, $x\neq 0$
({\cite[Proposition V.2.1]{16}}). In all questions considered in this note it can be supposed that $\|F\|\leq 1$ and $F$ is purely contractive.
The Operator Corona Problem is to find necessary and sufficient condition
for a function
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ to be left invertible, that is, to exist a function
$G\in H^\infty(\mathcal E_\ast\to\mathcal E)$ such that $G(z)F(z)=I_{\mathcal E}$ for all $z\in\mathbb D$.
If $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is left invertible, then there exists $\delta>0$
such that
\begin{equation}\label{1.1}\|F(z)x\|\geq \delta \|x\| \ \ \ \ \text{ for all } \ x\in\mathcal E, \ \ z\in\mathbb D. \end{equation}
It is easy to see that if $F$ satisfies \eqref{1.1} and $F=\Theta\Omega$ is the inner-outer factorization of $F$,
then the outer function $\Omega$ is invertible, and the inner function $\Theta$ satisfies \eqref{1.1}
(may be with another $\delta$).
The condition \eqref{1.1} is sufficient for left invertibility, if $\dim\mathcal E<\infty$ \cite{19}, but
is not sufficient in general \cite{20}, \cite{21}. Also, \eqref{1.1} is not sufficient under additional assumption
\begin{equation}\label{1.2}\dim\mathcal E_\ast\ominus F(z)\mathcal E=1 \ \ \text{ for all } \ z\in\mathbb D. \end{equation}
In \cite{23}, for every $\delta$, $0<\delta<1/3$, two functions $F_1$, $F_2\in H^\infty(\mathcal E\to\mathcal E_\ast)$
are constructed such that
\begin{equation}\label{1.3} \|x\|\geq\|F_k(z)x\|\geq \delta \|x\| \ \ \ \ \text{ for all } \ x\in\mathcal E, \ \ z\in\mathbb D,\end{equation}
\eqref{1.2} is fulfilled for $F_k$, $k=1,2$, $$F_1(\zeta)\mathcal E=\mathcal E_\ast \ \text{ for a.e. }\ \zeta\in\mathbb T,
\ \ \ \
\dim\mathcal E_\ast\ominus F_2(\zeta)\mathcal E=1 \ \text{ for a.e. }\ \zeta\in\mathbb T,$$
but $F_1$ and $F_2$ are not
left invertible. It is mentioned in \cite{23} that the method from \cite{20}, \cite{21} gives examples of such functions for
$\delta<1/\sqrt 2$, and a question was posed if for every $\delta$, $0<\delta<1$, there exists
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ such that $F$ satisfies \eqref{1.2} and \eqref{1.3}, and $F$ is not
left invertible.
In this note, it is shown that such function $F$ exists for every $0<\delta<1$, and
any from the following cases can be realized:
\begin{equation}\label{1.4}F(\zeta)\mathcal E=\mathcal E_\ast \ \ \text{ for a.e. } \ \zeta\in\mathbb T, \end{equation}
or
\begin{equation}\label{1.5}\dim\mathcal E_\ast\ominus F(\zeta)\mathcal E=1 \ \ \text{ for a.e. } \ \zeta\in\mathbb T, \end{equation}
or
\begin{equation}\label{1.6}\dim\mathcal E_\ast\ominus F(\zeta)\mathcal E=1 \ \ \text{ for a.e. } \ \zeta\in E, \ \text{ and }
\ \
F(\zeta)\mathcal E=\mathcal E_\ast \ \ \text{ for a.e. } \zeta\in\mathbb T\setminus E, \end{equation}
where
$E\subset\mathbb T$ is a closed set satisfying the Carleson condition with $0<m(E)<1$ (see Sec. 5
of this note where the definition is recalled).
Actually, not operator-valued functions, but contractions are constructed, and the required
functions are the characteristic functions of these contractions {\cite[Ch. VI]{16}}, see Sec. 3 of this note.
In \cite{22}, some sufficient conditions are given,
which imply the left invertibility of functions, in particular,
it is proved in \cite{22}, that if $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$
satisfies to \eqref{1.1} and
\begin{equation}\label{1.7}\sup_{z\in\mathbb D}\|I_{\mathcal E}-F(z)^\ast F(z)\|_{\frak S_1}<\infty,
\end{equation}
where $\frak S_1$ is the trace class of operators, then $F$ is left invertible.
In this note, it is shown that the condition \eqref{1.7} is necessary for left invertibility of $F$, if $F$ is inner
and $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$ for some $z\in\mathbb D$. Actually, an appropriate fact
is proved for contractions with such characteristic functions, and the statement on function follows from
the fact for contractions.
The function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ has a left scalar multiple if there exist
$G\in H^\infty(\mathcal E_\ast\to\mathcal E)$ and a function $\rho\in H^\infty$, where $H^\infty$ is the algebra of all
bounded analytic functions on $\mathbb D$, such that
$\rho(z)I_{\mathcal E}= G(z)F(z)$ for all $z\in\mathbb D$.
The left invertibility of $F$ means that $F$ has a scalar multiple, which is invertible in $H^\infty$.
Functions $F_1$ and $F_2$ from \cite{23} mentioned above do not have scalar multiple,
the proof is actually the same as the proof that $F_1$ and $F_2$ are not left invertible.
In this note, it is shown that the existence of the left scalar multiple of $F$ with \eqref{1.1} is not sufficient
for the left invertibility of $F$, even if $F$ is inner and $F$ satisfies \eqref{1.2}. In this case,
$I_{\mathcal E}-F(z)^\ast F(z)\in\frak S_1$ for every $z\in\mathbb D$, but
$\sup_{z\in\mathbb D}\|I_{\mathcal E}-F(z)^\ast F(z)\|_{\frak S_1}=\infty$. Again,
an appropriate
contraction is constructed, and $F$ is the characteristic function of this contraction.
We shall use the following notation: $\mathbb D$ is the open unit disk,
$\mathbb T$ is the unit circle, $m$ is the normalized Lebesgue measure on $\mathbb T$,
and $H^2$ is
the Hardy space in $\mathbb D$.
For a positive integer $n$, $1\leq n<\infty$, $H^2_n$ and $L^2_n$
are orthogonal sums of $n$ copies of spaces $H^2$ and $L^2 =L^2(\mathbb T,m)$,
respectively.
The unilateral shift $S_n$ and the bilateral shift $U_n$ of multiplicity $n$
are the operators of multiplication by the independent variable in spaces
$H^2_n$ and $L^2_n$, respectively. For a Borel set $\sigma\subset\mathbb T$, by
$U(\sigma)$ we denote the operator of multiplication by the
independent variable on the space $L^2(\sigma,m)$ of functions from $L^2$
that are equal to zero a.e. on $\mathbb T\setminus\sigma$.
For a Hilbert space $\mathcal H$, by $I_{\mathcal H}$ and $\mathbb O_{\mathcal H}$ the identity and the zero operators
acting on $\mathcal H$ are denoted, respectively.
Let $T$ and $R$ be operators on spaces $\mathcal H$ and $\mathcal K$,
respectively, and let $X:\mathcal H\to\mathcal K$ be a (linear, bounded) operator which
intertwines $T$ and $R$: $XT=RX$. If $X$ is unitary, then $T$ and $R$
are called {\it unitarily equivalent}, in notation:
$T\cong R$. If $X$ is invertible (the inverse $X^{-1}$ is bounded), then $T$ and $R$
are called {\it similar}, in notation: $T\approx R$.
If $X$ a quasiaffinity, that is, $\ker X=\{0\}$ and
$\operatorname{clos}X\mathcal H=\mathcal K$, then
$T$ is called a {\it quasiaffine transform} of $R$,
in notation: $T\prec R$. If $T\prec R$ and $R\prec T$,
then $T$ and $R$ are called {\it quasisimilar}, in notation: $T\sim R$.
Let $\mathcal H$ be a Hilbert space,
and let $T\colon\mathcal H\to\mathcal H$ be a (linear, bounded) operator. $T$ is called a {\it contraction}, if $\|T\|\leq 1$.
Let $T$ be a contraction
on a space $\mathcal H$. $T$ is {\it of class} $C_{1\cdot}$
($T\in C_{1\cdot}$), if $\lim_{n\to\infty}\|T^nx\|>0$ for
each $x\in \mathcal H$, $x\neq 0$, $T$ is {\it of class} $C_{0\cdot}$
($T\in C_{0\cdot}$), if
$\lim_{n\to\infty}\|T^nx\|=0$ for each $x\in\mathcal H$, and $T$ is
of class $C_{\cdot a}$, $a=0,1$, if
$T^\ast$ is of class $C_{a\cdot}$.
It is easy to see that if a contraction $T$
is a quasiaffine transform of an isometry, then $T$ is of class $C_{1\cdot}$,
and if $T$ is a quasiaffine transform of a unilateral shift,
then $T$ is of class $C_{10}$.
The paper is organized as follows.
In Sec. 2 and 3, the known facts about contractions, their relations to isometries,
and their characteristic functions are collected. In Sec. 4, the necessity of \eqref{1.7}
to the left invertibility of some operator-valued functions is proved. Sec. 5 is the main section
of this paper, where for any $\delta$, $0<\delta<1$, examples of subnormal contractions are constructed such that
their characteristic functions satisfy \eqref{1.2}, and \eqref{1.3} with $\delta$, and are not
left invertible.
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\section{Isometric and unitary asymptotes of contractions}
For a contraction $T$ the {\it isometric asymptote} $(X_{T,+},T_+^{(a)})$, and
the {\it unitary asymptote} $(X_T,T^{(a)})$ was defined,
see, for example, {\cite[Ch. IX.1]{16}}. There are some ways to construct
the isometric asymptote of the contraction, for our purpose,
it is convenient to use the following, see \cite{11}.
Let $(\cdot,\cdot)$ be the inner
product on the Hilbert space $\mathcal H$,
and let $T\colon\mathcal H\to\mathcal H$ be a contraction.
Define a new semi-inner product on
$\mathcal H$ by the formula $\langle x,y\rangle=\lim_{n\to\infty}(T^nx,T^ny)$,
where $x,y\in \mathcal H$. Set
$$\mathcal H_0=\mathcal H_{T,0}=\{x\in\mathcal H:\ \langle x, x\rangle=0\}.$$
Then the factor space $\mathcal H/\mathcal H_0$ with the inner product
$\langle x+\mathcal H_0, y+\mathcal H_0\rangle=\langle x,y\rangle$ will
be an inner product space.
Let $\mathcal H_+^{(a)}$ denote the resulting Hilbert space obtained
by completion, and let $X_{T,+}\colon\mathcal H\to\mathcal H_+^{(a)}$
be the natural imbedding,
$X_{T,+}x=x+\mathcal H_0$. Clearly, $X_{T,+}$ is a (linear, bounded) operator, and $\|X_{T,+}\|\leq 1$.
Clearly, $\langle Tx,Ty\rangle=\langle x,y\rangle$ for every $x,y\in \mathcal H$.
Therefore, $T_1: x+\mathcal H_0\mapsto Tx+\mathcal H_0$ is a well-defined isometry on
$\mathcal H/\mathcal H_0$. Denote by $T_+^{(a)}$ the continuous extension of $T_1$ to the space
$\mathcal H_+^{(a)}$. Clearly, $X_{T,+}T=T_+^{(a)}X_{T,+}$. The pair $(X_{T,+},T_+^{(a)})$ is called
the {\it isometric asymptote} of a contraction $T$. The operator $X_{T,+}$ is called the {\it canonical intertwining mapping}.
A contraction $T$ is similar
to an isometry $V$ if and only if $X_{T,+}$ is an invertible operator,
that is, $\ker X_{T,+}=\{0\}$ and $X_{T,+}\mathcal H= \mathcal H_+^{(a)}$, and in this case, $V\cong T_+^{(a)}$
(see {\cite[Theorem 1]{11}}). In particular, if $T$ is a contraction of class $C_{10}$, and $T_+^{(a)}$
is a unitary operator, then $T$ is not similar to an isometry.
Denote by $T^{(a)}$ the minimal unitary extension of $T_+^{(a)}$, by $\mathcal H^{(a)}\supset\mathcal H_+^{(a)}$
the space on which $T^{(a)}$ acts, and by $X_T$ the imbedding of $\mathcal H$ into $\mathcal H^{(a)}$.
Clearly, $X_TT=T^{(a)}X_T$ and $X_{T,+}x=X_Tx$ for every $x\in\mathcal H$.
The pair $(X_T,T^{(a)})$ is called the {\it unitary asymptote} $(X,T^{(a)})$ of a contraction $T$.
\section{Contractions and their characteristic functions}
All statement of this section are well-known and can be found in {\cite[Ch. VI]{16}}, see also
{\cite[Ch. C.1]{14}}.
Let $\mathcal H$ be a separable Hilbert space, and let $T\colon\mathcal H\to\mathcal H$ be
a contraction.
A contraction $T$ is called {\it completely nonunitary}, if $T$ has no invariant subspace such that
the restriction of $T$ on this subspace is unitary.
For a contraction $T$ put $\mathcal D_T=\operatorname{clos}(I_{\mathcal H}-T^\ast T)\mathcal H$.
It is easy to see that \begin{equation}\label{3.1}\text{ if } \ x\in\mathcal H\ominus\mathcal D_T, \ \
\text{ then } \ \ x=T^\ast Tx \ \ \text{ and } \ \ \|Tx\|=\|x\|. \end{equation}
Also, $T\mathcal D_T\subset\mathcal D_{T^\ast}$ and $T(\mathcal H\ominus\mathcal D_T)=
\mathcal H\ominus\mathcal D_{T^\ast}$ (see {\cite[Ch. I.3.1]{16}}), therefore,
\begin{equation}\label{3.2} \dim \mathcal D_{T^\ast}\ominus T\mathcal D_T = \dim\mathcal H\ominus T\mathcal H . \end{equation}
Since $I_{\mathcal H} - T^\ast T=(I_{\mathcal D_T} - T^\ast T|_{\mathcal D_T})\oplus\mathbb O_{\mathcal H\ominus\mathcal D_T}$,
\begin{equation}\label{3.3}\|I_{\mathcal H} - T^\ast T\|_{\frak S_1}=\|I_{\mathcal D_T} - T^\ast T|_{\mathcal D_T}\|_{\frak S_1}. \end{equation}
\begin{lemma}\label{lem3.1} Let $T\colon\mathcal H\to\mathcal H$ be a contraction, and let $0<\delta\leq 1$.
Then $\|Tx\|\geq\delta\|x\|$ for every $x\in\mathcal H$ if and only if
$\|Tx\|\geq\delta\|x\|$ for every $x\in\mathcal D_T$.\end{lemma}
\begin{proof} Indeed, it need to prove the ``if" part only. Let $x\in\mathcal D_T$, and let $y\in\mathcal H\ominus\mathcal D_T$.
Then, by \eqref{3.1}, $(Tx,Ty)=(x,T^\ast Ty)=(x,y)=0$ and $\|Ty\|=\|y\|\geq\delta\|y\|$.
Therefore, $\|T(x+y)\|^2 = \|Tx\|^2+\|Ty\|^2\geq\delta^2\|x\|^2+\delta^2\|y\|^2=\delta^2\|x+y\|^2$. \end{proof}
The characteristic function $\Theta_T$ of the contaction $T$ is the analytic
operator-valued function acting by the formula
\begin{equation}\label{3.4}
\Theta_T(z)=\big(-T+z (I-TT^\ast)^{1/2}(I-zT^\ast)^{-1}(I-T^\ast T)^{1/2}\big)|_{\mathcal D_T},
\ \ z\in\mathbb D. \end{equation}
For every $z\in\mathbb D$ the inclusion $\Theta_T(z)\mathcal D_T\subset\mathcal D_{T^\ast}$ holds,
the mapping $z\mapsto\Theta_T(z)$ is an analytic function from $\mathbb D$ to
the space of all (linear, bounded) operators from $\mathcal D_T$ to $\mathcal D_{T^\ast}$,
and $\|\Theta_T(z)\|\leq 1$ for every $z\in\mathbb D$.
That is, $\Theta_T\in H^\infty(\mathcal D_T\to\mathcal D_{T^\ast})$, and $\|\Theta_T\|\leq 1$. It is easy to see
that $\Theta_T$ is purely contractive.
Conversely, for every analytic operator-valued function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$
such that $\|F\|\leq 1$ and $F$ is purely contractive there exists a contraction $T$ such that $F=\Theta_T$ {\cite[Ch. VI]{16}}.
The following theorem was proved in \cite{15}, see also {\cite[C.1.5.5]{14}}.
\begin{theoremcite}\cite{15}\label{tha}
The contraction $T$ is similar to an isometry if and only if
$\Theta_T$ is left invertible.\end{theoremcite}
Let $T$ be a completely nonunitary contraction. Then $T$ is
of class $C_{1\cdot}$ if and only if $\Theta_T$ is $\ast$-outer,
and $T$ is of class $C_{\cdot 0}$,
if and only if $\Theta_T$ is inner {\cite[VI.3.5]{16}}.
Recall that the {\it multiplicity} of an operator is the minimum dimension
of its reproducing subspaces. An operator is called {\it cyclic} if its multiplicity is equal to $1$.
The following theorem was proved in \cite{7}, \cite{10}, \cite{17}, \cite{24}, \cite{25}.
\begin{theoremcite}\label{thb} Let $T$ be a contraction, and let
$1\leq n <\infty$. The following are equivalent:
$(1)$ $T\prec S_n$;
$(2)$ $T$ is of class $C_{10}$, $\dim\ker T^\ast =n$, and
$I-T^\ast T \in\frak S_1$;
$(3)$ $\Theta_T$ is an inner $\ast$-outer function, $\Theta_T$ has a left
scalar multiple, and $\dim\mathcal D_{T^\ast}\ominus\Theta_T(\lambda)\mathcal D_T=n$
for some $\lambda\in\mathbb D$.
Moreover, if $T$ is a contraction such that $T\prec S_n$, $1\leq n <\infty$, then the following are equivalent:
$(4)$ $T\sim S_n$;
$(5)$ multiplicity of $T$ is equal to $n$;
$(6)$ $\Theta_T$ has an outer left scalar multiple.\end{theoremcite}
{\bf Remark.} If $T$ is a contraction and $T\prec S_n$, $1\leq n <\infty$, then $b_\lambda(T)$ is a contraction and
$b_\lambda(T)\prec b_\lambda(S_n)\cong S_n$. Therefore, $I-b_\lambda(T)^\ast b_\lambda(T)\in\frak S_1$ for
every $\lambda\in\mathbb D$. Here $b_\lambda(T)=(T-\lambda)(I-\overline\lambda T)^{-1}$.
Let $T$ be a completely nonunitary contraction.
Put \begin{equation}\label{3.5}\Delta_\ast(\zeta)=(I_{\mathcal D_{T^\ast}}-
\Theta_T(\zeta)\Theta_T(\zeta)^\ast)^{1/2}, \ \ \zeta\in\mathbb T, \ \
\text{ and }\ \ \omega_T=\{\zeta\in\mathbb T:
\Delta_\ast(\zeta) \neq \mathbb O\}.\end{equation}
Then the unitary asymptote $T^{(a)}$ of a completely nonunitary contraction $T$
is unitarily equivalent to the operator of multiplication by the independent variable $\zeta$ on
$\operatorname{clos}\Delta_\ast L^2(\mathcal D_{T^\ast})$. In particular, $T^{(a)}$ is cyclic if and only if
\begin{equation}\label{3.6}\dim \Delta_\ast(\zeta) \mathcal D_{T^\ast}\leq 1 \ \ \text{ for a.e. } \ \zeta\in\mathbb T,
\end{equation} and in this case
$T^{(a)}\cong U(\omega_T)$ (see {\cite[Ch. IX.2]{16}}).
Also, if the function $\Theta_T$ is inner and \eqref{3.6} holds, then
\begin{equation}\label{3.7}\begin{aligned}\omega_T & = \{ \zeta\in\mathbb T: \
\dim \mathcal D_{T^\ast}\ominus\Theta(\zeta)\mathcal D_T = 1 \} \\
\text{ and } \ \ \mathbb T\setminus\omega_T & =\{ \zeta\in\mathbb T:
\ \Theta(\zeta)\mathcal D_T=\mathcal D_{T^\ast}\}.\end{aligned}\end{equation}
For $\lambda\in\mathbb D$ put $b_\lambda(z)=\frac{z-\lambda}{1-\overline\lambda z}$, $z\in\mathbb D$.
Then $b_\lambda(T)=(T-\lambda)(I-\overline\lambda T)^{-1}$ is a contraction. For every $\lambda\in\mathbb D$ there exists
unitary operators
$$V_\lambda\colon\mathcal D_T\to\mathcal D_{b_\lambda(T)} \ \ \ \text{ and } \ \ \
V_{\lambda\ast}\colon\mathcal D_{b_\lambda(T)^\ast}\to\mathcal D_{T^\ast}$$
such that
\begin{equation}\label{3.8} V_{\lambda\ast}\Theta_{b_\lambda(T)}(z)V_\lambda=
\Theta_T(b_{-\lambda}(z)). \end{equation}
Setting $z=0$ in \eqref{3.4} and \eqref{3.8}, we conclude that
\begin{equation}\label{3.9}\Theta_T(\lambda)=-V_{\lambda\ast}b_\lambda(T)V_\lambda \ \ \text{ for every } \
\lambda\in\mathbb D \end{equation}
({\cite[Ch. VI.1.3]{16}}).
The following lemma is a straightforward consequence of \eqref{3.2}, \eqref{3.3}, \eqref{3.9},
and Lemma \ref{lem3.1}.
\begin{lemma}\label{lem3.2} Suppose $T\colon\mathcal H\to\mathcal H$ is a completely
nonunitary contraction.
$\ \ \text{\rm (i)}$ Let $0<\delta\leq1$. Then $\|\Theta_T(\lambda)x\|\geq\delta\|x\|$
for every $x\in\mathcal D_T$, $\lambda\in\mathbb D$,
if and only if $\|b_\lambda(T)x\|\geq\delta\|x\|$ for every
$x\in\mathcal H$, $\lambda\in\mathbb D$.
$\ \text{\rm (ii)}$ $\dim\mathcal D_{T^\ast}\ominus\Theta_T(\lambda)\mathcal D_T=
\dim\mathcal H\ominus b_\lambda(T)\mathcal H$
for every $\lambda\in\mathbb D$.
$\text{\rm (iii)}$ $\|I_{\mathcal D_T}-
\Theta_T^\ast(\lambda)\Theta_T(\lambda)\|_{\frak S_1} =
\|I_{\mathcal H}-b_\lambda(T)^\ast b_\lambda(T) \|_{\frak S_1}$
for every $\lambda\in\mathbb D$.\end{lemma}
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\section{On contractions similar to an isometry}
The following theorem is actually proved in {\cite[Theorem 2.1]{7}} (see also enlarged version on arXiv).
\begin{theorem}\label{th4.1} \cite{7} Suppose $T$ is a contraction
with finite multiplicity,
and $T$ is similar to an isometry.
Then \begin{equation}\label{4.1}\sup_{\lambda\in\mathbb D}\|I-b_\lambda(T)^\ast b_\lambda(T)\|_{\frak S_1}
<\infty. \end{equation} \end{theorem}
\begin{corollary}\label{cor4.2} Suppose $\mathcal E$, $\mathcal E_\ast$ are Hilbert spaces,
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is an inner function, and
$\dim \mathcal E_\ast\ominus F(\lambda)\mathcal E<\infty$ for some $\lambda \in\mathbb D$.
If $F$ is left invertible, then $F$ satisfies \eqref{1.7}.\end{corollary}
\begin{proof} Let $\mathcal H$ be a Hilbert space, and let $T\colon\mathcal H\to\mathcal H$ be
a contraction such that $\Theta_T=F$, where $\Theta_T$ is the characteristic function
of $T$ (see {\cite[Ch. VI.3]{16}}). Since $F$ is inner, $T\in C_{\cdot 0}$, see {\cite[VI.3.5]{16}}.
By Lemma \ref{lem3.2}(ii),
$$\dim \mathcal H\ominus b_\lambda(T)\mathcal H = \dim \mathcal E_\ast\ominus F(\lambda)\mathcal E<\infty.$$
Now suppose that $F$ is left invertible. Then, by Theorem \ref{tha}, there exist a Hilbert space
$\mathcal K$ and an isometry $V\colon\mathcal K\to\mathcal K$ such that $T\approx V$. Since $T\in C_{\cdot 0}$,
$V$ is a unilateral shift, and the multiplicity of $V$ is equal to
$$\dim \mathcal K\ominus b_\lambda(V)\mathcal K = \dim \mathcal H\ominus b_\lambda(T)\mathcal H <\infty.$$
By Theorem \ref{th4.1}, $T$ satisfies \eqref{4.1}. By Lemma \ref{lem3.2}(iii), $F$ satisfies \eqref{1.7}. \end{proof}
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0.55.4
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\section{Subnormal contractions}
Operators that are considered in this sections are subnormal ones, and are studied by
many authors, the reader can consult with the book \cite{6}.
Let $\nu$ be
a positive finite Borel measure on the closed unit disk $\overline{\mathbb D}$. Denote by
$P^2(\nu)$ the closure of analytic polynomials in $L^2(\nu)$, and by $S_\nu$
the operator of multiplication by the independent variable in $P^2(\nu)$, i.e.
\begin{align*} & S_\nu: P^2(\nu)\to P^2(\nu), \\ & (S_\nu f)(z)=zf(z) \text{ for a.e. } z\in \overline{\mathbb D}
\ \text{ with respect to } \nu , \ \ f\in P^2(\nu).\end{align*}
Clearly, $S_\nu$ is a contraction.
Recall that $m$ is the Lebesgue measure on $\mathbb T$. If $\nu=m$, then $S_\nu$
is the unilateral shift of multiplicity 1, it is denoted by $S$ in this section.
The following lemma is a straightforward consequence of the construction of the
isometric asymptote of a contraction from \cite{11}, see Sec. 2 of this paper, so its proof is omitted.
\begin{lemma}\label{lem5.1} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, $\mathcal H=P^2(\nu)$, and $T=S_\nu$.
Then $\mathcal H_{0,T}=\{f\in P^2(\nu): \ f=0 \ \text {a.e. on}\ \ \mathbb T
\text { with respect to}\ \nu\}$, $\mathcal H_+^{(a)}=P^2(\nu|_{\mathbb T})$,
$$X_{T,+}\colon P^2(\nu)\to P^2(\nu|_{\mathbb T}), \ \ \ X_{T,+}f=f|_{\mathbb T}, \ \ f\in P^2(\nu),$$
is the natural imbedding, and $T_+^{(a)}=S_{\nu|_{\mathbb T}}$.\end{lemma}
The proof of the following lemma is obvious and omitted.
\begin{lemma}\label{lem5.2} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, and $f\in P^2(\nu)$. Then there exists $\lambda\in \mathbb D$
such that $\|b_\lambda f\|=\|f\|$ if and only if $f(z)=0$ for a.e. $z\in\mathbb D$
with respect to $\nu$.\end{lemma}
\begin{corollary}\label{cor5.3} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, and $\lambda\in \mathbb D$. Then
$$ P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)}\subset
\{f\in P^2(\nu):\ f(z)=0 \ \text{ for a.e. }\ z\in\mathbb D
\ \text{ with respect to } \nu\}$$
and
$$P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)^\ast}\subset
\{f\in P^2(\nu):\ f(z)=0 \ \text{ for a.e. }\ z\in\mathbb D
\ \text{ with respect to } \nu\}.$$\end{corollary}
\begin{proof} Let $P_+:L^2(\nu)\to P^2(\nu)$ be the orthogonal projection.
It is easy to see that $(b_\lambda(S_\nu)f)(z)=b_\lambda(z)f(z)$ for a.e. $z\in \overline{\mathbb D}$
with respect to $\nu$, and $b_\lambda(S_\nu)^\ast f=P_+(\overline b_\lambda f)$, $f\in P^2(\nu)$.
If $f\in P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)}$, then, by \eqref{3.1}, $\|f\|=\| b_\lambda f\|$,
and, by Lemma \ref{lem5.2}, $f(z)=0$ for a.e. $z\in\mathbb D$ with respect to $\nu$.
If $f\in P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)^\ast}$, then, by \eqref{3.1},
$\|f\|=\| P_+(\overline b_\lambda f)\|\leq\|\overline b_\lambda f\|=\|b_\lambda f\|$,
and, by Lemma \ref{lem5.2}, $f(z)=0$ for a.e. $z\in\mathbb D$ with respect to $\nu$. \end{proof}
Denote by $m_2$ the normalized Lebesgue measure on the unit disk $\mathbb D$, for $-1<\alpha<\infty$
put $\text{\rm d}A_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha\text{\rm d}m_2(z)$. It is well known that
the Bergman space $P^2(A_\alpha)$ has the following properties:
$f\in P^2(A_\alpha)$ if and only $f$ is an analytic function in $\mathbb D$ and $f\in L^2(A_\alpha)$,
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha)$ for every $z\in\mathbb D$,
and there exists a constant $C_\alpha>0$ (which depends on $\alpha$) such that
\begin{equation}\label{5.1} |f(z)|\leq C_\alpha\frac{\|f\|_{P^2(A_\alpha)}}{(1-|z|^2)^{1+\alpha/2}},
\ \ \ z\in\mathbb D, \end{equation}
(see, for example, {\cite[Sec. 1.1 and 1.2]{9}}). It is easy to see that
$S_{A_\alpha}\in C_{00}$.
\begin{lemma}\label{lem5.4} {\cite[Lemma 4.2]{3}} Let $-1<\alpha<\infty$.
Then for every $f\in P^2(A_\alpha)$ and $\lambda\in\mathbb D$
$$\int_{\mathbb D}|b_\lambda f|^2\text{\rm d}A_\alpha\geq
\frac{1}{\alpha+2}\int_{\mathbb D}|f|^2\text{\rm d}A_\alpha.$$\end{lemma}
\begin{corollary}\label{cor5.5} Let $-1<\alpha<\infty$, and let $\mu$ be a positive finite Borel measure on $\mathbb T$.
Then for every $f\in P^2(A_\alpha+\mu)$ and $\lambda\in\mathbb D$
$$\int_{\overline{\mathbb D}}|b_\lambda f|^2\text{\rm d}(A_\alpha+\mu)\geq
\frac{1}{\alpha+2}\int_{\overline{\mathbb D}}|f|^2\text{\rm d}(A_\alpha+\mu).$$\end{corollary}
\begin{proof} Clearly, $P^2(A_\alpha+\mu)\subset P^2(A_\alpha)$. Let
$f\in P^2(A_\alpha+\mu)$, and let $\lambda\in\mathbb D$. We have
\begin{align*}\int_{\overline{\mathbb D}} |b_\lambda & f|^2\text{\rm d}(A_\alpha+\mu) =
\int_{\mathbb D}|b_\lambda f|^2\text{\rm d}A_\alpha +
\int_{\mathbb T}|b_\lambda f|^2\text{\rm d}\mu \\ &\geq
\frac{1}{\alpha+2}\int_{\mathbb D}|f|^2\text{\rm d}A_\alpha +
\int_{\mathbb T}|f|^2\text{\rm d}\mu
\geq\frac{1}{\alpha+2}\int_{\overline{\mathbb D}}|f|^2\text{\rm d}(A_\alpha+\mu),\end{align*}
because of $|b_\lambda|=1$ on $\mathbb T$ and $1> 1/(\alpha+2)$ for $-1<\alpha<\infty$. \end{proof}
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Recall the following definition.
{\bf Definition.} Let $E$ be a closed subset of $\mathbb T$, and let $\{J_k\}_k$
be the collection of open arcs of $\mathbb T$ such that $J_k\cap J_\ell=\emptyset$
for $k\neq\ell$ and
$\mathbb T=E\cup\bigcup_kJ_k$. The set $E$ satisfies the {\it Carleson condition}
if $\sum_km(J_k)\log m(J_k)>-\infty$.
Let $w\in L^1(\mathbb T,m)$, $w\geq 0$ a.e. on $\mathbb T$. Then $P^2(wm)=L^2(wm)$ if and only if
$\log w\not\in L^1(\mathbb T,m)$, and then $S_{wm}\cong U(\sigma)$, where $\sigma\subset\mathbb T$ is
a measurable set such that $wm$ and $m|_\sigma$ are mutually absolutely continuous.
If $ \log w\in L^1(\mathbb T,m)$, then there exists an outer function $\psi\in H^2$ such that
$|\psi|^2=w$ a.e. on $\mathbb T$. Then
\begin{equation}\label{5.2} \begin{aligned} P^2(wm) & =\frac{H^2}{\psi}=\Big\{\frac{h}{\psi}: \ h\in H^2\Big\},
\ \ \Big\|\frac{h}{\psi}\Big\|_{P^2(wm)}=\|h\|_{H^2}, \ h\in H^2,\\
\text{ and } \ \ S_{wm} & \cong S \end{aligned} \end{equation}
(see, for example, {\cite[Ch. III.12]{6}} or {\cite[A.4.1.5]{14}}).
In Theorems \ref{th5.6} and \ref{th5.7}, we consider nontangential boundary values of functions from
$P^2(\mu)$ for some measures $\mu$.
Nontangential boundary values of functions from
$P^t(\mu)$ with $1\leq t<\infty$ are considered in \cite{4} in relation to another questions, see also references therein,
especially \cite{1}, \cite{12}, \cite{13}, \cite{18}, and \cite{2}. In Theorems \ref{th5.6} and \ref{th5.7} we formulate particular cases of these results in the form convenient to our purpose.
Theorems \ref{th5.6} and the main part of Theorem \ref{th5.7} were proved in {\cite[Sec. 2]{8}} for $\alpha=0$,
but the proofs are the same in
the case of $-1<\alpha\leq 0$ (because the estimate \eqref{5.1} involves the estimate
$|f(z)|\leq C_\alpha\frac{\|f\|_{P^2(A_\alpha)}}{1-|z|^2}$ for
$-1<\alpha\leq 0$, which is used in {\cite[Sec. 2]{8}}), therefore,
the proofs of Theorem \ref{th5.6} and of the main part of Theorem \ref{th5.7} are omitted.
In addition, to prove Theorems \ref{th5.6} and \ref{th5.7}, one needs to apply the notion of isometric asymptote (see Sec. 2 of this paper and references therein).
\begin{theorem}\label{th5.6} \cite{8} Let $-1<\alpha\leq 0$, and let $E\subset\mathbb T$ be a closed set
such that $0<m(E)<1$ and $E$ satisfies the Carleson condition. Then
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha+m|_E)$ for every $z\in\mathbb D$.
Furthermore,
for
$f\in P^2(A_\alpha+m|_E)$ the restriction $f|_{\mathbb D}$ is analytic on $\mathbb D$,
$f|_{\mathbb D}$ has nontangential boundary values
a.e. on $E$ with respect to $m$, which coincide with $f|_E$.
Therefore, $S_{A_\alpha+m|_E}\in C_{10}$.
Also, $I-S_{A_\alpha+m|_E}^\ast S_{A_\alpha+m|_E}$ is compact, and
$(S_{A_\alpha+m|_E})^{(a)}_+=U(E)$. Thus, $S_{A_\alpha+m|_E}$
is not similar to an isometry.\end{theorem}
\begin{theorem}\label{th5.7} \cite{8} Let $-1<\alpha\leq 0$, and let
$w\in L^1(\mathbb T,m)$.
Suppose that for every closed arc $J\subset\mathbb T\setminus\{1\}$
there exist two constants
$0<c_J<C_J<\infty$ such that $c_J\leq w\leq C_J$ a.e. on $J$
(with respect to $m$). Then
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha+wm)$
for every $z\in\mathbb D$. Furthermore, for
$f\in P^2(A_\alpha+wm)$ the restriction $f|_{\mathbb D}$ is analytic on
$\mathbb D$, $f|_{\mathbb D}$ has nontangential boundary values
a.e. on $\mathbb T$ with respect to $m$, which coincide with $f|_{\mathbb T}$.
Therefore, $S_{A_\alpha+wm}\in C_{10}$.
Also, $I-S_{A_\alpha+wm}^\ast S_{A_\alpha+wm}$ is compact, $(S_{A_\alpha+wm})^{(a)}_+=S_{wm}$,
and the canonical mapping which intertwines $S_{A_\alpha+wm}$ with $S_{wm}$
is the natural imbedding
$$P^2(A_\alpha+wm)\to P^2(wm), \ \ f\mapsto f|_{\mathbb T}, \ f\in P^2(A_\alpha+wm).$$
Therefore,
$\ \ \text{\rm (i)}$ $S_{A_\alpha+wm}\sim S$ if and only if $\log w\in L^1(\mathbb T,m)$;
$\ \text{\rm (ii)}$ $S_{A_\alpha+wm}$ is similar to an isometry if and only if $S_{A_\alpha+wm}\approx S$;
$\text{\rm (iii)}$ $S_{A_\alpha+wm}\approx S$ if and only if
$\log w\in L^1(\mathbb T,m)$ and for every $h\in H^2$ there exists $f \in P^2(A_\alpha+wm)$
such that $f|_{\mathbb T}=h/\psi$ a.e. on $\mathbb T$ (with respect to $m$), where $\psi\in H^2$ is
an outer function such that
$|\psi|^2=w$ a.e. on $\mathbb T$. \end{theorem}
{\bf Remark.} In the conditions of Theorem \ref{th5.7}, let $h,\psi\in H^2$, $\psi(z)\neq 0$ for every $z\in\mathbb D$,
$f\in P^2(A_\alpha+wm)$, and $f|_{\mathbb T} =h/\psi$ a.e. on $\mathbb T$ (with respect to $m$). Then
$f(z)=h(z)/\psi(z)$ for every $z\in\mathbb D$. Indeed, set $g(z)=h(z)/\psi(z)$, $z\in\mathbb D$. Then
$g$ is a function analytic on $\mathbb D$, and $g$ has nontangential boundary values $h(\zeta)/\psi(\zeta)$
for a.e. $\zeta\in\mathbb T$. Then $f-g$ is a function analytic on $\mathbb D$, and $f-g$ has zero nontangential
boundary values a.e. on $\mathbb T$. By Privalov's theorem (see, for example, {\cite[Theorem 8.1]{5}}),
$f(z)=g(z)$ for every $z\in\mathbb D$. Therefore, if the conditions (iii) of Theorem \ref{th5.7} are fulfilled,
then $ P^2(A_\alpha+wm)=H^2/\psi$ as the set, and the norms on these spaces are equivalent.
{\it Proof of Theorem \ref{th5.7}.} The main part of Theorem \ref{th5.7} is proved in {\cite[Sec. 2]{8}}.
Let $X$ be the imbedding,
$$X\colon P^2(A_\alpha+wm)\to P^2(wm), \ \ Xf=f|_{\mathbb T}, \ \ f\in P^2(A_\alpha+wm).$$
Then $XS_{A_\alpha+wm}=S_{wm}X$. Since $S_{A_\alpha+wm}\in C_{1\cdot}$, $\ker X=\{0\}$,
therefore, $X$ is a quasiaffinity which realizes the relation $S_{A_\alpha+wm}\prec S_{wm}$.
If $\log w\in L^1(\mathbb T,m)$, then $S_{wm}\cong S$, therefore, $S_{A_\alpha+wm}\prec S$.
Since $S_{A_\alpha+wm}$ is a cyclic contraction, $S_{A_\alpha+wm}\sim S$ by Theorem \ref{thb}(5).
The ``if" part of (i) is proved.
The assumptions of the ``if" part of (iii) mean that $X P^2(A_\alpha+wm) = P^2(wm)$, see \eqref{5.2}.
Thus, $X$ realizes the relation $S_{A_\alpha+wm}\approx S_{wm}$, and, since $S_{wm}\cong S$,
the relation $S_{A_\alpha+wm}\approx S$ is proved.
Now suppose that $S_{A_\alpha+wm}\approx V$, where $V$ is an isometry. Then, by {\cite[Theorem 1]{11}}
(see Sec. 2 of this paper), $X P^2(A_\alpha+wm) = P^2(wm)$ and $V\cong S_{wm}$. Since
$S_{A_\alpha+wm}\in C_{10}$, $S_{wm}\in C_{10}$. By {\cite[A.4.1.5]{14}} (see the description of $S_{wm}$
before \eqref{5.2} in this paper), $\log w\in L^1(\mathbb T,m)$ and $S_{wm}\cong S$. The parts (ii) and (iii)
are proved.
Now suppose that $S_{A_\alpha+wm}\sim S$. By {\cite[Theorem 1]{11}}, see also {\cite[Ch. IX.1]{16}}, there exists
an operator $Y\colon P^2(wm)\to H^2$ such that $YS_{wm}=SY$ and $\operatorname{clos}Y P^2(wm)= H^2$.
If $\log w\not\in L^1(\mathbb T,m)$, then $S_{wm}$ is unitary, and from the relations $Y^\ast S^\ast=S_{wm}^\ast Y^\ast$
and $\ker Y^\ast=\{0\}$ we conclude that $S^\ast\in C_{1\cdot}$, a contradiction. Therefore,
$\log w\in L^1(\mathbb T,m)$. The ``only if" part of (i) is proved. \qed
The following lemma is a variant of {\cite[Theorem 1.7]{9}}.
\begin{lemma}\label{lem5.8} Let $-1<\alpha<\infty$, and let $\beta\in\mathbb R$.
Put $\varphi_\beta(z)=1/(1-z)^\beta$, $z\in\mathbb D$. Then
$\varphi_\beta\in H^2$ if and only if $\beta<1/2$, and
$\varphi_\beta\in P^2(A_\alpha)$ if and only if $\beta<1+\alpha/2$.\end{lemma}
\begin{proof} Put $v_n=2(\alpha+1)\int_0^1r^{2n+1}(1-r^2)^\alpha\text{\rm d}r$, $n\geq 0$.
Then $v_n=\frac{n!\Gamma(\alpha+2)}{\Gamma(\alpha+n+2)}$, where $\Gamma$ is the Gamma function,
and
for every function $f$ analytic on $\mathbb D$
$$\int_{\mathbb D}|f|^2\text{\rm d} A_\alpha=\sum_{n=0}^\infty |\widehat f(n)|^2 v_n.$$
If $\beta\leq 0$, then $\varphi_\beta\in P^2(A_\alpha)$ for every $\alpha$, $-1<\alpha<\infty$. Suppose $\beta>0$.
Since $\widehat \varphi_{\beta}(n)=\frac{\Gamma(\beta+n)}{n!\Gamma(\beta)}$, $n\geq 0$, we have that
$\varphi_\beta\in P^2(A_\alpha)$ if and only if the series
$\sum_{n=0}^\infty
\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}$ converges. By Stirling's formula,
$$\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}\sim (n+1)^{2\beta-\alpha-3} \ \ \text { as } \ n\to\infty.$$
Therefore, the series
$\sum_{n=0}^\infty
\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}$ converges if and only if $\beta<1+\alpha/2$.
The first statement of the lemma can be proved similarly. \end{proof}
\begin{lemma}\label{lem5.9} Let $-1<\alpha\leq 0$, and let $\beta<-1-\alpha$. Then $S_{A_\alpha+|\varphi_\beta| m}\sim S$,
but $S_{A_\alpha+|\varphi_\beta| m}$ is not similar to an isometry.\end{lemma}
\begin{proof}
By Theorem \ref{th5.7}(i),
$S_{A_\alpha+|\varphi_\beta|m}\sim S$. Put $\psi=\varphi_{\beta/2}$. Clearly, $|\psi|^2=|\varphi_\beta|$.
By \eqref{5.2}, $P^2(|\varphi_\beta|m)=H^2/\psi$. By Theorem \ref{th5.7}(iii), if $S_{A_\alpha+|\varphi_\beta|m}$
is similar to an isometry, then $H^2/\psi = P^2(A_\alpha+|\varphi_\beta|m)\subset P^2(A_\alpha)$.
Take $\gamma$, $1+\alpha/2+\beta/2\leq\gamma<1/2$. Put $h=\varphi_\gamma$. Then $h\in H^2$,
and $h/\psi=\varphi_{\gamma-\beta/2}\not\in P^2(A_\alpha)$
by Lemma \ref{lem5.8}. Therefore, $S_{A_\alpha+|\varphi_\beta|m}$ is not similar to an isomertry. \end{proof}
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\begin{lemma}\label{lem5.9} Let $-1<\alpha\leq 0$, and let $\beta<-1-\alpha$. Then $S_{A_\alpha+|\varphi_\beta| m}\sim S$,
but $S_{A_\alpha+|\varphi_\beta| m}$ is not similar to an isometry.\end{lemma}
\begin{proof}
By Theorem \ref{th5.7}(i),
$S_{A_\alpha+|\varphi_\beta|m}\sim S$. Put $\psi=\varphi_{\beta/2}$. Clearly, $|\psi|^2=|\varphi_\beta|$.
By \eqref{5.2}, $P^2(|\varphi_\beta|m)=H^2/\psi$. By Theorem \ref{th5.7}(iii), if $S_{A_\alpha+|\varphi_\beta|m}$
is similar to an isometry, then $H^2/\psi = P^2(A_\alpha+|\varphi_\beta|m)\subset P^2(A_\alpha)$.
Take $\gamma$, $1+\alpha/2+\beta/2\leq\gamma<1/2$. Put $h=\varphi_\gamma$. Then $h\in H^2$,
and $h/\psi=\varphi_{\gamma-\beta/2}\not\in P^2(A_\alpha)$
by Lemma \ref{lem5.8}. Therefore, $S_{A_\alpha+|\varphi_\beta|m}$ is not similar to an isomertry. \end{proof}
\begin{corollary}\label{cor5.10} Let $0<\delta<1$, and let $E\subset\mathbb T$ be a closed set
satisfying the Carleson condition and such that $0<m(E)<1$. Then there exist operator-valued
inner functions $F_k$,
such that $F_k$ satisfy $\eqref{1.2}$, and $\eqref{1.3}$ with $\delta$, and $F_k$ are not
left invertible,
$k=1,2,3$. Also, $F_1$, $F_2$, $F_3$ satisfy $\eqref{1.4}$, $\eqref{1.6}$, $\eqref{1.5}$, respectively,
$F_3$ has an outer left scalar multiple, and $I-F_3(z)^\ast F_3(z)\in\frak S_1$ for every $z\in\mathbb D$.
\end{corollary}
{\bf Remark.} Since $F_3$ is not left invertible, $F_3$ does not satisfy \eqref{1.7}, see \cite{22}.
{\it Proof of Corollary \ref{cor5.10}.} Put $\alpha=1/\max(\delta^2, 1/2)-2$, then $-1<\alpha\leq 0$.
Take $\beta<-1-\alpha$. Put
$$\mathcal H_1=P^2(A_\alpha), \ \ \mathcal H_2=P^2(A_\alpha+m|_E), \ \ \mathcal H_3=P^2(A_\alpha+|\varphi_\beta|m), $$
$$ T_1=S_{A_\alpha}, \ \ T_2=S_{A_\alpha+m|_E}, \ \ T_3=S_{A_\alpha+|\varphi_\beta|m}.$$
Clearly, $T_k$ are cyclic contractions, $k=1,2,3$, $T_1\in C_{00}$, and $T_k\in C_{10}$ by
Theorems \ref{th5.6} and \ref{th5.7},
$k=2,3$. By Corollary \ref{cor5.5}, \begin{equation}\label{5.3} \|b_\lambda(T_k)f\|^2\geq \delta^2\|f\|^2 \ \text{ for every } \ \lambda\in\mathbb D, \
\ f\in\mathcal H_k, \ \ k=1,2,3. \end{equation}
By Theorems \ref{th5.6} and \ref{th5.7}, the functionals
$$ f\mapsto f(z), \ \ \mathcal H_k\to\mathbb C,$$
are bounded for every $z\in\mathbb D$, $k=1,2,3$.
Therefore, \begin{equation}\label{5.4} \dim \mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k =1 \ \text{ for every } \ \lambda\in\mathbb D, \
\ \ k=1,2,3. \end{equation}
Indeed, let $k$ be fixed, and let $\lambda\in\mathbb D$. Then there exists $g_\lambda\in \mathcal H_k$ such that
$f(\lambda)=(f,g_\lambda)$ for every $f\in \mathcal H_k$. Since $(b_\lambda(T_k)f)(z)=b_\lambda(z)f(z)$ for every
$z\in\mathbb D$, $f\in \mathcal H_k$, it is clear that $g_\lambda\in\mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k$.
Since $T_k$ is cyclic, $T_k-\lambda I$ is cyclic, too, therefore,
$\dim \mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k = \dim \mathcal H_k\ominus (T_k-\lambda I)\mathcal H_k\leq 1$.
The equality \eqref{5.4} is proved.
Now find the unitary asymptotes of $T_k$, $k=1,2,3$, see Sec. 2 of this paper and references therein.
Since $T_1\in C_{00}$, $T_1^{(a)}=\mathbb O$. By Theorem \ref{th5.6}, $T_2^{(a)}=U(E)$. By Lemma \ref{lem5.9},
$T_3\sim S$, therefore, $T_3^{(a)}=U(\mathbb T)$, the bilateral shift of multiplicity 1.
Since $T^{(a)}\cong U(\omega_T)$ for every cyclic completely nonunitary contraction $T$, where
$\omega_T$ is defined in \eqref{3.5}, we conclude that
\begin{equation}\label{5.5} \omega_{T_1}=\emptyset, \ \ \ \omega_{T_2}=E, \ \ \ \omega_{T_3}=\mathbb T. \end{equation}
Also, $T_1$ is not similar to an isometry, because $T_1\in C_{00}$, and $T_2$ and $T_3$ are
not similar to an isometry by Theorem \ref{th5.6} and Lemma \ref{lem5.9}, respectively.
Now put $F_k=\Theta_{T_k}$, that is, $F_k$ is the characteristic function of the contraction $T_k$,
$k=1,2,3$, see Sec. 3 of this paper and references therein. Since $T_k\in C_{\cdot 0}$, $F_k$ are inner.
By \eqref{5.3} and Lemma \ref{lem3.2}(i), $F_k$ satisfy \eqref{1.3} with $\delta$. By \eqref{5.4} and
Lemma \ref{lem3.2}(ii), $F_k$ satisfy \eqref{1.2}. $F_k$ are not left invertible, because of $T_k$ are not
similar to an isometry, see Theorem \ref{tha}. $F_1$, $F_2$, $F_3$ satisfy \eqref{1.4}, \eqref{1.6}, \eqref{1.5}, respectively,
because of \eqref{3.7} and \eqref{5.5}. Since $T_3\sim S$ (by Lemma \ref{lem5.9}), $F_3$ has an outer
left scalar multiple
by Theorem \ref{thb}(6), and $I-F_3(z)^\ast F_3(z)\in\frak S_1$ for every $z\in\mathbb D$
by Lemma \ref{lem3.2}(iii) and Theorem \ref{thb}(2).
\qed
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\begin{document}
\title{Minimum number of additive tuples in groups of prime order}
\begin{abstract} For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_0,\dots,A_k\subseteq\Z{p}$ of sizes $a_0,\dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo~$p$. By applying basic Fourier analysis, we show for Bajnok's problem that if $p\geqslant 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\varepsilonquiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.
\varepsilonnd{abstract}
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\section{Introduction}
Let $\Gamma$ be a given finite Abelian group, with the group operation written additively.
For $A_0,\dots,A_k\subseteq\Gamma$, let $s(A_0,\dots,A_k)$ be the number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots+x_k$. If $A_0=\dots=A_k:=A$, then we use the shorthand $s_k(A):=S(A_0,\dots,A_k)$. For example, $s_2(A)$ is the number of \varepsilonmph{Schur triples} in $A$, that is, ordered triples $(x_0,x_1,x_2)\in A^3$ with $x_0=x_1+x_2$.
For integers $n\geqslant m\geqslant 0$, let $[m,n]:=\{m,m+1,\dots,n\}$ and
$[n]:=[0,n-1]=\{0,\dots,n-1\}$.
For a sequence $a_0,\dots,a_k\in [\,|\Gamma| +1\,] = \lbrace 0,1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}, |\Gamma|\rbrace$, let $s(a_0,\dots,a_k;\Gamma)$ be the minimum of $s(A_0,\dots,A_k)$ over subsets $A_0,\dots,A_k\subseteq \Gamma$ of sizes $a_0,\dots,a_k$ respectively. Additionally, for $a\in [0,p]$, let $s_k(a;\Gamma)$ be the minimum of $s_k(A)$ over all $a$-sets $A\subseteq \Gamma$.
The question of finding the maximal size of a sum-free subset of $\Gamma$ (i.e.\ the maximum $a$ such that $s_2(a;\Gamma)=0$) originated in a paper of Erd\H os~\cite{Erdos65} in 1965 and took 40 years before it was resolved in full generality by Green and Ruzsa~\cite{GreenRuzsa05}.
In this paper, we are interested in the case where $p$ is a fixed prime and the underlying group $\Gamma$ is taken to be $\Z p$, the cyclic group of order $p$,
which we identify with the additive group of residues modulo $p$ (also using the multiplicative structure on it when this is useful).
Lev~\cite{Lev01duke} solved the problem of finding $s_k(a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k;\Z p)$, where $p$ is prime (in the equivalent guise of considering solutions to $x_1+\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} + x_k=0$).\footnote{We learned of Lev's work after the publication of this paper. For completeness, we still provide a proof of Theorem~\ref{th:main} in Section~\ref{knot=1}, which is essentially the same as the original proof of Lev's more general result, and which was rediscovered in~\cite{SamotijSudakov16pmcps}.}
For $I\subseteq\Z p$ and $x,y\in\Z p$, write $x\cdot I+y:=\{x\cdot z+y: z\in I\}$.
\begin{theorem}\cite{Lev01duke} \label{th:main} For arbitrary $k\geqslant 1$ and $a_0,\dots,a_k\in [0,p]$, there is $t\in\Z p$ such that
$$
s(a_0,\dots,a_k;\Z p)=s([a_0]+t,[a_1],\dots,[a_k]; \Z p).\qed
$$
\varepsilonnd{theorem}
Huczynska, Mullen and Yucas~\cite{HuczynskaMullenYucas09jcta} found a new proof of the $s_2(a; \Z p)$-problem, while also addressing some extensions. Samotij and Sudakov~\cite{SamotijSudakov16pmcps} rediscovered Lev's proof of the $s_2(a ; \Z p)$-problem and showed that, when $s_2(a ,\Z p)>0$, then the $a$-sets that achieve the minimum
are exactly those of the form $\xi\cdot I$ with $\xi\in\Z{p}\setminus\{0\}$, where $I$ consists of the residues modulo $p$ of $a$ integers closest to $\frac{p-1}2\in\I Z$. Each such set is an arithmetic progression; its difference can be any non-zero value but the initial element has to be carefully chosen.
(By an \varepsilonmph{$m$-term arithmetic progression} (or \varepsilonmph{$m$-AP} for short) we mean a set of the form $\{x,x+d,\dots,x+(m-1)d\}$ for some $x,d\in\Z p$ with $d\not=0$. We call $d$ the \varepsilonmph{difference}.)
Samotij and Sudakov~\cite{SamotijSudakov16pmcps} also solved the $s_2(a)$-problem for various groups $\Gamma$.
Bajnok~\cite[Problem~G.48]{Bajnok18acmrp} suggested the more general problem of considering $s_k(a;\Gamma)$.
This is wide open in full generality.
This paper concentrates on the case $\Gamma = \Z p$, for $p$ prime, and the sets which attain equality in Theorem~\ref{th:main}.
In particular, we write $s(a_0,\dots,a_k):= s(a_0,\dots,a_k;\Z p)$ and $s_k(a):=s_k(a;\Z p)$. Since the case $p=2$ is trivial, let us assume that $p\geqslant 3$.
Since
\begin{equation}\label{eq:equiv}
s(A_0,\dots,A_k)=s(\xi\cdot A_0+\varepsilonta_0,\dots,\xi\cdot A_k+\varepsilonta_k),\quad\mbox{for $\xi\not=0$ and $\varepsilonta_0=\varepsilonta_1+\dots+\varepsilonta_k$},
\varepsilonnd{equation}
Theorem~\ref{th:main} shows that, for any difference $d$, there is at least one extremal configuration consisting of $k+1$ arithmetic progressions with the same difference $d$.
In particular, if $a_0=\dots=a_k=:a$, then one extremal configuration consists of $A_1=\dots=A_k=[a]$ and $A_0=[t,t+a-1]$ for some $t\in\Z p$.
Given this, one can write down some formulas for $s(a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k)$ in terms of $a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k$ involving summation (based on~\varepsilonqref{eq:s} or a version of~\varepsilonqref{eq:sk(A)}) but there does not seem to be a closed form in general.
If $k\not\varepsilonquiv1\pmod p$, then by taking $\xi:=1$, $\varepsilonta_1:=\dots:=\varepsilonta_k:=-t(k-1)^{-1}$, and $\varepsilonta_0:=-kt(k-1)^{-1}$ in~\varepsilonqref{eq:equiv}, we can get another extremal configuration where all sets are the same: $A_0+\varepsilonta_0=\dots=A_k+\varepsilonta_k$. Thus Theorem~\ref{th:main} directly implies the following corollary.
\begin{corollary}\label{cr:main} For every $k\geqslant 2$ with $k\not\varepsilonquiv1\pmod p$ and $a\in [0,p]$, there is $t\in\Z p$ such that $s_k(a)=s_k([t,t+a-1])$.\qed\varepsilonnd{corollary}
Unfortunately, if $k\geqslant 3$, then there may be sets $A$ different from APs that attain equality in Corollary~\ref{cr:main} with $s_k(|A|)>0$ (which is in contrast to the case $k=2$). For example, our (non-exhaustive) search showed that this happens already for $p=17$, when
$$
s_3(14)=2255=s_3([-1,12])=s_3([6,18]\cup\{3\}).
$$
Also, already the case $k=2$ of the more general Theorem~\ref{th:main} exhibits extra solutions. Of course, by analysing the proof of Theorem~\ref{th:main} or Corollary~\ref{cr:main} one can write a necessary and sufficient condition for the cases of equality. We do this in Section~\ref{knot=1}; in some cases this condition can be simplified.
The first main result of this paper is
to describe the extremal sets for Corollary~\ref{cr:main} when $k \not\varepsilonquiv 1 \pmod p$ is sufficiently large.
The proof uses basic Fourier analysis on $\Z p$.
\begin{theorem}\label{th:knot1} Let a prime $p\geqslant 7$ and an integer $a\in [3,p-3]$ be fixed, and let $k\not\varepsilonquiv1\pmod p$ be sufficiently large. Then
there exists $t \in \Z p$ for which the only $s_k(a)$-extremal sets are $\xi\cdot[t,t+a-1]$ for all non-zero $\xi \in \Z p$.
\varepsilonnd{theorem}
\begin{problem} Find a `good' description of all extremal families for Corollary~\ref{cr:main} (or perhaps Theorem~\ref{th:main}) for $k\geqslant 3$.\varepsilonnd{problem}
While Corollary~\ref{cr:main} provides an example of an $s_k(a)$-extremal set for $k\not\varepsilonquiv1\pmod p$, the case $k\varepsilonquiv1\pmod p$ of the $s_k(a)$-problem turns out to be somewhat special. Here, translating a set $A$ has no effect on the quantity $s_k(A)$. More generally, let $\mathcal{A}$ be the group of all invertible affine transformations of $\Z p$, that is, it consists of maps $x\mapsto \xi\cdot x+\varepsilonta$, $x\in\Z p$, for $\xi,\varepsilonta\in \Z p$ with $\xi\not=0$. Then \begin{equation}\label{eq:equiv1}
s_k(\alpha(A))=s_k(A),\quad \mbox{for every $k\varepsilonquiv1\!\!\pmod p$\ \ and\ \ $\alpha\in\mathcal{A}$}.
\varepsilonnd{equation}
Let us call two subsets $A,B\subseteq \Z p$ \varepsilonmph{(affine) equivalent} if there is $\alpha\in \mathcal{A}$ with $\alpha(A)=B$. By~\varepsilonqref{eq:equiv1}, we need to consider sets only up to this equivalence.
Trivially, any two subsets of $\Z p$ of size $a$ are equivalent if $a \leqslantq 2$ or $a \geqslantq p-2$.
Our second main result, is to describe the extremal sets when $k \varepsilonquiv 1 \pmod p$ is sufficiently large, again using Fourier analysis on $\Z p$.
\begin{theorem}\label{th:k1} Let a prime $p\geqslant 7$ and an integer $a\in [3,p-3]$ be fixed, and let $k\varepsilonquiv1\pmod p$ be sufficiently large. Then the following statements hold for the $s_k(a)$-problem.
\begin{enumerate}
\item If $a$ and $k$ are both even, then $[a]$ is the unique (up to affine equivalence) extremal set.
\item If at least one of $a$ and $k$ is odd, define $I':=[a-1]\cup\{a\}=\{0,\dots,a-2,a\}$. Then
\begin{enumerate}
\item $s_k(a)<s_k([a])$ for all large $k$;
\item $I'$ is the unique extremal set for infinitely many $k$;
\item $s_k(a)<s_k(I')$ for infinitely many $k$, provided there are at least three non-equivalent $a$-subsets of $\Z p$.
\varepsilonnd{enumerate}
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
It is not hard to see that there are at least three non-equivalent $a$-subsets of $\Z p$ if and only if $p\geqslant 13$ and $a\in [3,p-3]$, or $p\geqslant 11$ and $a\in [4,p-4]$. Thus Theorem~\ref{th:k1} characterises pairs $(p,a)$ for which there exists an $a$-subset $A$ which is $s_k(a)$-extremal for \varepsilonmph{all} large $k\varepsilonquiv1\pmod p$.
\begin{corollary} Let $p$ be a prime and $a\in[0,p]$. There is an $a$-subset $A\subseteq \Z p$ with $s_k(A)=s_k(a)$ for all large $k\varepsilonquiv1\pmod p$ if and only if $a\leqslant 2$, or $a\geqslant p-2$, or $p\in\{7,11\}$ and $a=3$.\qed
\varepsilonnd{corollary}
As is often the case in mathematics, a new result leads to further open problems.
\begin{problem} Given $a\in[3,p-3]$, find a `good' description of all $a$-subsets of $\Z p$ that are $s_k(a)$-extremal for at least one (resp.\ infinitely many) values of $k\varepsilonquiv1\pmod p$.\varepsilonnd{problem}
\begin{problem} Is it true that for every $a\in[3,p-3]$ there is $k_0$ such that for all $k\geqslant k_0$ with $k\varepsilonquiv 1\pmod p$, any two $s_k(a)$-extremal sets are affine equivalent?\varepsilonnd{problem}
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\section{Proof of Theorem~\ref{th:main}}\label{knot=1}
For completeness, here we prove Theorem~\ref{th:main}, which is a special case of Theorem~1 in~\cite{Lev01duke}.
Let $A_1,\dots,A_k$ be subsets of $\Z p$. Define $\sigma(x;A_1,\dots,A_k)$ as the number of $k$-tuples $(x_1,\dots,x_k)\in A_1\times\dots\times A_k$ with $x=x_1+\dots+x_k$. Also, for an integer $r\geqslant 0$, let
\begin{eqnarray*}
N_r(A_1,\dots,A_k)&:=&\{x\in\Z p: \sigma(x;A_1,\dots,A_k)\geqslant r\},\\
n_r(A_1,\dots,A_k)&:=&|N_r(A_1,\dots,A_k)|.
\varepsilonnd{eqnarray*}
These notions are related to our problem because of the following easy identity:
\begin{equation}\label{eq:s}
s(A_0,\dots,A_k)=\sum_{r=1}^\infty |A_0\cap N_r(A_1,\dots,A_k)|.
\varepsilonnd{equation}
Let an \varepsilonmph{interval} mean an arithmetic progression with difference $1$, i.e.\ a subset $I$ of $\Z p$ of form $\{x,x+1,\dots,x+y\}$. Its \varepsilonmph{centre} is $x+y/2\in \I Z_p$; it is unique if $I$ is \varepsilonmph{proper} (that is, $0<|I|<p$).
Note the following easy properties of the sets $N_r$:
\begin{enumerate}
\item These sets are nested:
\begin{equation}\label{eq:nested}
N_0(A_1,\dots,A_k)=\Z p\supseteq N_1(A_1,\dots,A_k)\supseteq N_2(A_1,\dots,A_k)\supseteq \dots
\varepsilonnd{equation}
\item If each $A_i$ is an interval with centre $c_i$, then $N_r(A_1,\dots,A_k)$ is an interval with centre $c_1+\dots+c_k$.
\varepsilonnd{enumerate}
We will also need the following result of Pollard~\cite[Theorem~1]{Pollard75}.
\begin{theorem}\label{th:Pollard} Let $p$ be a prime, $k\geqslant 1$, and $A_1,\dots,A_k$ be subsets of $\Z{p}$ of sizes $a_1,\dots,a_k$. Then for every integer $r\geqslant 1$, we have
$$
\sum_{i=1}^r n_i(A_1,\dots,A_k)\geqslant \sum_{i=1}^r n_i([a_1],\dots,[a_k]).\qed
$$
\varepsilonnd{theorem}
\bpf[Proof of Theorem~\ref{th:main}] Let $A_0,\dots,A_k$ be some extremal sets for the $s(a_0,\dots,a_k)$-problem. We can assume that $0<a_0<p$, because
$s(A_0,\dots,A_k)$ is $0$ if $a_0=0$ and $\prod_{i=1}^ka_i$
if $a_0=p$, regardless of the choice of the sets $A_i$.
Since $n_0([a_1],\dots,[a_k])=p>p-a_0$ while $n_r([a_1],\dots,[a_k])=0<p-a_0$ when, for example, $r>\prod_{i=1}^{k-1} a_i$, there is a (unique) integer $r_0\geqslant 0$ such that
\begin{eqnarray}
n_{r}([a_1],\dots,[a_k])&>&p-a_0,\quad\mbox{all $r\in [0,r_0]$,}\label{eq:r01}\\
n_{r}([a_1],\dots,[a_k])&\leqslant &p-a_0,\quad\mbox{all integers $r\geqslant r_0+1$.}\label{eq:r02}
\varepsilonnd{eqnarray}
The nested intervals $N_1([a_1],\dots,[a_k])\supseteq N_2([a_1],\dots,[a_k])\supseteq\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}$ have the same centre $c:=((a_1-1)+\dots+(a_k-1))/2$. Thus there is a translation $I:=[a_0]+t$ of $[a_0]$, with $t$ independent of $r$, which has as small as possible intersection with each $N_r$-interval above given their sizes, that is,
\begin{equation}\label{eq:intersection}
|I\cap N_r([a_1],\dots,[a_k])|=\max\{\,0,\,n_r([a_1],\dots,[a_k])+a_0-p\,\},\quad \mbox{for all $r\in\I N$}.
\varepsilonnd{equation}
This and Pollard's theorem give the following chain of inequalities:
\begin{eqnarray*}
s(A_0,\dots,A_k)&\stackrel{\varepsilonqref{eq:s}}{=}& \sum_{i=1}^\infty |A_0\cap N_i(A_1,\dots,A_k)|\\
&\geqslant & \sum_{i=1}^{r_0} |A_0\cap N_i(A_1,\dots,A_k)|\\
&\geqslant & \sum_{i=1}^{r_0} (n_i(A_1,\dots,A_k)+a_0-p)\\
&\stackrel{\mathrm{Thm~\ref{th:Pollard}}}{\geqslant} & \sum_{i=1}^{r_0} (n_i([a_1],\dots,[a_k])+a_0-p)\\
&\stackrel{\varepsilonqref{eq:r01}-\varepsilonqref{eq:r02}}{=} & \sum_{i=1}^{\infty} \max\{\,0,\, n_i([a_1],\dots,[a_k])+a_0-p\,\}\\
&\stackrel{\varepsilonqref{eq:intersection}}=& \sum_{i=1}^{\infty}|I\cap N_i([a_1],\dots,[a_k])|\\
&\stackrel{\varepsilonqref{eq:s}}=& s(I,[a_1],\dots,[a_k]),
\varepsilonnd{eqnarray*}
giving the required.\varepsilonpf
Let us write a necessary and sufficient condition for equality in Theorem~\ref{th:main} in the case $a_0,\dots,a_k\in [1,p-1]$. Let $r_0\geqslant 0$ be defined by \varepsilonqref{eq:r01}--\varepsilonqref{eq:r02}. Then, by~\varepsilonqref{eq:nested}, a sequence $A_0,\dots,A_k\subseteq \Z p$ of sets of sizes respectively $a_0,\dots,a_k$ is extremal if and only if
\begin{eqnarray}
A_0\cap N_{r_0+1}(A_1,\dots,A_k)&=&\varepsilonmptyset,\label{eq:empty}\\
A_0\cup N_{r_0}(A_1,\dots,A_k)&=& \Z p,\label{eq:whole}\\
\sum_{i=1}^{r_0} n_i(A_1,\dots,A_k)&=& \sum_{i=1}^{r_0} n_i([a_1],\dots,[a_k]).\label{eq:PollEq}
\varepsilonnd{eqnarray}
Let us now concentrate on the case $k=2$, trying to simplify the above condition. We can assume that no $a_i$ is equal to 0 or $p$ (otherwise the choice of the other two sets has no effect on $s(A_0,A_1,A_2)$ and every triple of sets of sizes $a_0$, $a_1$ and $a_2$ is extremal). Also, as in~\cite{SamotijSudakov16pmcps}, let us exclude the case $s(a_0,a_1,a_2)=0$, as then there are in general many extremal configurations. Note that $s(a_0,a_1,a_2)=0$ if and only if $r_0=0$; also,
by the Cauchy-Davenport theorem (the special case $k=2$ and $r=1$ of Theorem~\ref{th:Pollard}), this is equivalent to $a_1+a_2-1\leqslant p-a_0$.
Assume by symmetry that $a_1\leqslant a_2$.
Note that~\varepsilonqref{eq:r01} implies that $r_0\leqslant a_1$.
The condition in~\varepsilonqref{eq:PollEq} states that we have equality in Pollard's theorem. A result of Nazarewicz, O'Brien, O'Neill and Staples~\cite[Theorem~3]{NazarewiczObrienOneillStaples07} characterises when this happens (for $k=2$), which in our notation is the following.
\begin{theorem}\label{th:NazarewiczObrienOneillStaples07} For $k=2$ and $1\leqslant r_0\leqslant a_1\leqslant a_2<p$, we have equality in~\varepsilonqref{eq:PollEq} if and only if at least one of the following conditions holds:
\begin{enumerate}
\item\label{it:1} $r_0=a_1$,
\item\label{it:2} $a_1+a_2\geqslant p+r_0$,
\item\label{it:3} $a_1=a_2=r_0+1$ and $A_2=g-A_1$ for some $g\in \Z{p}$,
\item\label{it:4} $A_1$ and $A_2$ are arithmetic progressions with the same difference.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
Let us try to write more explicitly each of these four cases, when combined with~\varepsilonqref{eq:empty} and~\varepsilonqref{eq:whole}.
First, consider the case $r_0=a_1$. We have $N_{a_1}([a_1],[a_2])=[a_1-1,a_2-1]$ and thus $n_{a_1}([a_1],[a_2])=a_2-a_1+1>p-a_0$, that is, $a_2-a_1\geqslant p-a_0$. The condition~\varepsilonqref{eq:empty} holds automatically since $N_i(A_1,A_2)=\varepsilonmptyset$ whenever $i>|A_1|$. The other condition~\varepsilonqref{eq:whole} may be satisfied even when none of the sets $A_i$ is an arithmetic progression (for example, take $p=13$, $A_1=\{0,1,3\}$, $A_2=\{0,2,3,5,6,7,9,10\}$ and let $A_0$ be the complement of $N_3(A_1,A_2)=\{3,6,10\}$). We do not see any better characterisation here, apart from stating that~\varepsilonqref{eq:whole} holds.
Next, suppose that $a_1+a_2\geqslant p+r_0$. Then, for any two sets $A_1$ and $A_2$ of sizes $a_1$ and $a_2$, we have $N_{r_0}(A_1,A_2)=\Z p$; thus~\varepsilonqref{eq:whole} holds automatically. Similarly to the previous case, there does not seem to be a nice characterisation of~\varepsilonqref{eq:empty}. For example,~\varepsilonqref{eq:empty} may hold even when none of the sets $A_i$ is an AP: e.g.\ let $p=11$, $A_1=A_2=\{0,1,2,3,4,5,7\}$, and let $A_0=\{0,2,10\}$ be the complement of $N_4(A_1,A_2)=\{1,3,4,5,6,7,8,9\}$ (here $r_0=3$).
\comment{Let us verify that indeed $r_0=3$. Indeed, $n_3([7],[7])=11$ by above while $N_4([7],[7])=[3,9]$ has $7\leqslant 11-a_0=8$ elements.
}
Next, suppose that we are in the third case. The primality of $p$ implies that $g\in\Z p$ satisfying $A_2=g-A_1$ is unique and thus $N_{r_0+1}(A_1,A_2)=\{g\}$. Therefore~\varepsilonqref{eq:empty} is equivalent to $A_0\not\ni g$. Also, note that if $I_1$ and $I_2$ are intervals of size $r_0+1$, then $n_{r_0}(I_1,I_2)=3$. By the definition of $r_0$, we have
$p-2\leqslant a_0\leqslant p-1$.
Thus we can choose any integer $r_0\in [1,p-2]$ and $(r_0+1)$-sets $A_2=g-A_1$, and then let $A_0$ be obtained from $\Z p$ by removing $g$ and at most one further element of $N_{r_0}(A_1,A_2)$. Here, $A_0$ is always an AP (as a subset of $\Z p$ of size $a_0\geqslant p-2$) but $A_1$ and $A_2$ need not be.
Finally, let us show that if $A_1$ and $A_2$ are arithmetic progressions with the same difference $d$ and we are not in Case~1 nor~2 of Theorem~\ref{th:NazarewiczObrienOneillStaples07}, then $A_0$ is also an arithmetic progression whose difference is~$d$. By~\varepsilonqref{eq:equiv}, it is enough to prove this when $A_1=[a_1]$ and $A_2=[a_2]$ (and $d=1$).
Since $a_1+a_2\leqslant p-1+r_0$ and $r_0+1\leqslant a_1\leqslant a_2$, we have that
\begin{eqnarray*} N_{r_0}(A_1,A_2)&=&[r_0-1,a_1+a_2-r_0-1]\\
N_{r_0+1}(A_1,A_2)&=& [r_0,a_1+a_2-r_0-2]
\varepsilonnd{eqnarray*}
have sizes respectively $a_1+a_2-2r_0+1<p$ and $a_1+a_2-2r_0-1>0$. We see that $N_{r_0+1}(A_1,A_2)$ is obtained from the proper interval $N_{r_0}(A_1,A_2)$ by removing its two endpoints. Thus $A_0$, which is sandwiched between the complements of these two intervals by~\varepsilonqref{eq:empty}--\varepsilonqref{eq:whole}, must be an interval too. (And, conversely, every such triple of intervals is extremal.)
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\section{The proof of Theorems~\ref{th:knot1} and~\ref{th:k1}}
Let us recall the basic definitions and facts of Fourier analysis on $\Z p$. For a more detailed treatment of this case, see e.g.~\cite[Chapter~2]{Terras99faofg}.
Write $\omega := e^{2\pi i /p}$ for the $p^{\mathrm{th}}$ root of unity.
Given a function $f : \Z p \rightarrow \mathbb{C}$, we define its \varepsilonmph{Fourier transform} to be the function $\fourier{f}:\Z p\to \mathbb{C}$ given by
$$
\fourier{f}(\gamma) := \sum_{x=0}^{p-1} f(x)\, \omega^{-x\gamma}, \qquad\text{for } \gamma \in \Z p.
$$
Parseval's identity states that
\begin{equation}\label{eq:Parseval}
\sum_{x=0}^{p-1} f(x)\,\overline{g(x)} = \frac{1}{p}\sum_{\gamma=0}^{p-1} \fourier{f}(\gamma)\,\overline{\fourier{g}(\gamma)}.
\varepsilonnd{equation}
The \varepsilonmph{convolution} of two functions $f,g : \Z p \rightarrow \mathbb{C}$ is given by
$$
(f * g)(x) := \sum_{y=0}^{p-1}f(y)\,g(x-y).
$$
It is not hard to show that the Fourier transform of a convolution equals the product of Fourier transforms, i.e.
\begin{equation}\label{convolution}
\fourier{f_1 * \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} * f_k} = \fourier{f_1} \cdot\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} \cdot \fourier{f_k}.
\varepsilonnd{equation}
We write $f^{*k}$ for the convolution of $f$ with itself $k$ times. (So, for example, $f^{* 2} = f * f$.)
Denote by $\mathbbm{1}_A$ the \varepsilonmph{indicator function} of $A \subseteq \Z p$ which assumes value 1 on $A$ and $0$ on $\Z p\setminus A$. We will call $\fourier{{\mathbbm{1}}}_A(0)=|A|$ the \varepsilonmph{trivial Fourier coefficient of $A$}.
Since the Fourier transform behaves very nicely with respect to convolution, it is not surprising that our parameter of interest, $s_k(A)$, can be written as a simple function of the Fourier coefficients of $\mathbbm{1}_A$.
Indeed,
let $ A \subseteq \mathbb{Z}_p$ and $x \in \Z p$.
Then the number of tuples $(a_1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k) \in A^k$ such that $a_1+\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}+a_k=x$ (which is $\sigma(x;A,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},A)$ in the notation of Section~\ref{knot=1}) is precisely $\mathbbm{1}^{* k}_A(x)$. The function $s_k(A)$ counts such a tuple if and only if its sum $x$ also lies in $A$.
Thus,
\begin{equation}\label{eq:sk(A)}
s_k(A) = \sum_{x = 0}^{p-1}\mathbbm{1}^{* k}_A(x)\, \mathbbm{1}_A(x) \stackrel{(\ref{eq:Parseval})}= \frac{1}{p}\sum_{\gamma=0}^{p-1}\fourier{\mathbbm{1}^{* k}_A}(\gamma)\,\overline{\fourier{\mathbbm{1}_A}(\gamma)} \stackrel{(\ref{convolution})}{=} \frac{1}{p} \sum_{\gamma=0}^{p-1} \leqslantft(\fourier{\mathbbm{1}_A}(\gamma)\right)^k\, \overline{\fourier{\mathbbm{1}_A}(\gamma)}.
\varepsilonnd{equation}
Since every set $A \subseteq \Z p$ of size $a$ has the same trivial Fourier coefficient (namely $\fourier{\mathbbm{1}_A}(0)=a$), let us re-write~\varepsilonqref{eq:sk(A)} as
\beq{eq:SkF1}
p s_k(A)-a^{k+1}= \sum_{\gamma=1}^{p-1} (\fourier{\mathbbm{1}_A}(\gamma))^k\,
\O{\fourier{\mathbbm{1}_A}(\gamma)} =: F(A).
\varepsiloneq
Thus we need to minimise $F(A)$ (which is a real number for any $A$) over $a$-subsets $A\subseteq\Z p$.
To do this when $k$ is sufficiently large, we will consider the largest in absolute value non-trivial Fourier coefficient $\fourier{\mathbbm{1}_{A}}(\gamma)$ of an $a$-subset $A$. Indeed, the term $(\fourier{\mathbbm{1}_A}(\gamma))^k\overline{\fourier{\mathbbm{1}_A}(\gamma)}$ will dominate $F(A)$, so if it has strictly negative real part, then $F(A)<F(B)$ for all $a$-subsets~$B\subseteq \Z p$ with $\max_{\delta\not=0}|\fourier{\mathbbm{1}_B}(\delta)|<|\fourier{\mathbbm{1}_A}(\gamma)|$.
Given $a \in [p-1]$, let
$$
I := [a]=\lbrace 0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a-1\rbrace\quad\text{and}\quad I' := [a-1]\cup\lbrace a \rbrace = \lbrace a,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a-2,a\rbrace.
$$
In order to prove Theorems~\ref{th:knot1} and~\ref{th:k1}, we will make some preliminary observations about these special sets.
The set of $a$-subsets which are affine equivalent to $I$ is precisely the set of $a$-APs.
Next we will show that
\begin{equation}\label{skI}
F(I) = 2\sum_{\gamma=1}^{(p-1)/2} (-1)^{\gamma(a-1)(k-1)} \leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1}\quad\text{if }k \varepsilonquiv 1 \pmod p.
\varepsilonnd{equation}
Note that $(-1)^{\gamma(a-1)(k-1)}$ equals $(-1)^\gamma$ if both $a,k$ are even and 1 otherwise.
To see~\varepsilonqref{skI}, let $\gamma \in \lbrace 1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},\frac{p-1}{2}\rbrace$ and write $\fourier{\mathbbm{1}_I}(\gamma) = re^{\theta i}$ for some $r >0$ and $0 \leqslantq \theta < 2\pi$.
Then $\theta$ is the midpoint of $0,-2\pi \gamma/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}, -2(a-1)\gamma\pi/p$,~i.e.
$
\theta = -\pi(a-1)\gamma/p
$.
Choose $s \in \mathbb{N}$ such that $k = sp+1$.
Then
\begin{equation}\label{termx}
(\fourier{\mathbbm{1}_I}(\gamma))^k \overline{\fourier{\mathbbm{1}_I}(\gamma)} = \leqslantft(r e^{-\pi i (a-1)\gamma/p}\right)^k r e^{\pi i (a-1)\gamma/p} = r^{k+1} e^{-\pi i(a-1)\gamma s},
\varepsilonnd{equation}
and $e^{-\pi i(a-1)s}$ equals $1$ if $(a-1)s$ is even, and $-1$ if $(a-1)s$ is odd.
Note that, since $p$ is an odd prime, $(a-1)s$ is odd if and only if $a$ and $k$ are both even.
So~(\ref{termx}) is real, and the fact that $\fourier{\mathbbm{1}_I}(p-\gamma) = \overline{\fourier{\mathbbm{1}_I}(\gamma)}$ implies that the corresponding term for $p-\gamma$ is the same as for $\gamma$.
This gives~\varepsilonqref{skI}.
A very similar calculation to~(\ref{termx}) shows that
\begin{equation}\label{skI2}
F(I+t) = \sum_{\gamma=1}^{p-1} e^{-\pi i (2t+a-1)(k-1)\gamma/p}|\fourier{\mathbbm{1}_{I+t}}(\gamma)|^{k+1}\quad\text{for all }k \geqslantq 3.
\varepsilonnd{equation}
Given $r>0$ and $0 \leqslantq \theta < 2\pi$, we write $\arg(re^{\theta i}) := \theta$.
\begin{proposition}\label{angleprop}
Suppose that $p \geqslantq 7$ is prime and $a \in [3,p-3]$.
Then
$\arg\leqslantft(\fourier{\mathbbm{1}_{I'}}(1)\right)$ is not an integer multiple of $\pi/p$.
\varepsilonnd{proposition}
\bpf
Since $\fourier{\mathbbm{1}_A}(\gamma)=-\fourier{\mathbbm{1}_{\Z p\setminus A}}(\gamma)$ for all $A \subseteq \Z p$ and non-zero $\gamma \in \Z p$, we may assume without loss of generality that $a \leqslantq p-a$. Since $p$ is odd, we have $a \leqslantq (p-1)/2$.
Suppose first that $a$ is odd.
Let $m := (a-1)/2$. Then $m \in [1,\frac{p-3}{4}]$.
Observe that translating any $A \subseteq \Z p$ changes the arguments of its Fourier coefficients by an integer multiple of $2\pi/p$.
So, for convenience of angle calculations, here we may redefine $I := [-m,m]$ and $I' := \lbrace -m-1\rbrace\cup[-m+1,m]$.
Also let $I^- := [-m+1,m-1]$, which is non-empty.
The argument of $\fourier{\mathbbm{1}_{I^-}}(1)$ is $0$.
Further, $\fourier{\mathbbm{1}_{I'}}(1) = \fourier{\mathbbm{1}_{I^-}}(1) + \omega^{m+1}+\omega^{-m}$.
Since $\omega^{m+1},\omega^{-m}$ lie on the unit circle, the argument of $\omega^{m+1}+\omega^{-m}$ is either $\pi/p$ or $\pi+\pi/p$. But the bounds on $m$ imply that it has positive real part, so $\arg(\omega^{m+1}+\omega^{-m})=\pi/p$.
By looking at the non-degenerate parallelogram in the complex plane with vertices $0,\fourier{\mathbbm{1}_{I^-}}(1),\omega^{m+1}+\omega^{-m},\fourier{\mathbbm{1}_{I'}}(1)$, we see that the argument of $\fourier{\mathbbm{1}_{I'}}(1)$ lies strictly between that of $\fourier{\mathbbm{1}_{I^-}}(1)$ and $\omega^{m+1}+\omega^{-m}$, i.e.~strictly between $0$ and $\pi/p$, giving the required.
\begin{figure}[h]\label{figure}
\centering
\begin{tikzpicture}
\clip (-3.1,-3.1) rectangle (9,3.1);
\draw[thick,gray,dotted] (-3.5,0) -- (9,0);
\foreach \x in {0,15.65217,31.30435,...,350}
{
\draw[gray!50] (0,0) -- (\x:3);
}
\draw[black,thick] (0,0) circle (3cm);
\draw (0,0) node[circle,inner sep=2,fill = black,label=left:{$0$}] (0) {};
\draw (46.9563:3) node[circle,inner sep=2,fill = black,label=right:{$\omega^{m+1}$}] (m+1) {};
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=right:{$\omega^{-m}$}] (-m) {};
\draw[red,very thick] (31.3043:2.7) arc (31.3043:-31.3043:2.7);
\draw[] (2.5,-0.75) node[draw=none,label=above:{\textcolor{red}{$I$}}] () {};
\draw[red,thick] (31.3043:2.6) -- (31.3043:2.8);
\draw[red,thick] (-30.9:2.6) -- (-30.9:2.8);
\begin{scope}[shift=(m+1)]
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=above:{~~~~$\omega^{m+1}+\omega^{-m}$}] (pip) {};
\draw[gray!50] (0,0) -- (pip);
\varepsilonnd{scope}
\draw (46.9563:3) node[circle,inner sep=2,fill = black] (m+1) {};
\draw[black,thick] (0,0) -- (m+1);
\draw[black,thick] (0,0) -- (-m);
\draw[black,thick] (0,0) -- (pip);
\draw (0:4.5) node[circle,inner sep=2, fill=black,label=below:{$\fourier{\mathbbm{1}_{I^-}}(1)$}] (I-) {};
\draw[thick] (0,0) -- (I-);
\begin{scope}[shift=(I-)]
\draw[] (7.8261:4) node[circle,inner sep=2, fill=black,label=above:{$\fourier{\mathbbm{1}_{I'}}(1)$}] (pipshift) {};
\draw[gray!50] (0,0) -- (pipshift);
\varepsilonnd{scope}
\draw (0,0) -- (pipshift);
\draw[gray!50] (pipshift) -- (pip);
\draw[] (I-) node[circle,inner sep=2, fill=black] () {};
\varepsilonnd{tikzpicture}
\varepsilonnd{figure}
| 3,943 | 25,722 |
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0.56.4
|
\begin{proposition}\label{angleprop}
Suppose that $p \geqslantq 7$ is prime and $a \in [3,p-3]$.
Then
$\arg\leqslantft(\fourier{\mathbbm{1}_{I'}}(1)\right)$ is not an integer multiple of $\pi/p$.
\varepsilonnd{proposition}
\bpf
Since $\fourier{\mathbbm{1}_A}(\gamma)=-\fourier{\mathbbm{1}_{\Z p\setminus A}}(\gamma)$ for all $A \subseteq \Z p$ and non-zero $\gamma \in \Z p$, we may assume without loss of generality that $a \leqslantq p-a$. Since $p$ is odd, we have $a \leqslantq (p-1)/2$.
Suppose first that $a$ is odd.
Let $m := (a-1)/2$. Then $m \in [1,\frac{p-3}{4}]$.
Observe that translating any $A \subseteq \Z p$ changes the arguments of its Fourier coefficients by an integer multiple of $2\pi/p$.
So, for convenience of angle calculations, here we may redefine $I := [-m,m]$ and $I' := \lbrace -m-1\rbrace\cup[-m+1,m]$.
Also let $I^- := [-m+1,m-1]$, which is non-empty.
The argument of $\fourier{\mathbbm{1}_{I^-}}(1)$ is $0$.
Further, $\fourier{\mathbbm{1}_{I'}}(1) = \fourier{\mathbbm{1}_{I^-}}(1) + \omega^{m+1}+\omega^{-m}$.
Since $\omega^{m+1},\omega^{-m}$ lie on the unit circle, the argument of $\omega^{m+1}+\omega^{-m}$ is either $\pi/p$ or $\pi+\pi/p$. But the bounds on $m$ imply that it has positive real part, so $\arg(\omega^{m+1}+\omega^{-m})=\pi/p$.
By looking at the non-degenerate parallelogram in the complex plane with vertices $0,\fourier{\mathbbm{1}_{I^-}}(1),\omega^{m+1}+\omega^{-m},\fourier{\mathbbm{1}_{I'}}(1)$, we see that the argument of $\fourier{\mathbbm{1}_{I'}}(1)$ lies strictly between that of $\fourier{\mathbbm{1}_{I^-}}(1)$ and $\omega^{m+1}+\omega^{-m}$, i.e.~strictly between $0$ and $\pi/p$, giving the required.
\begin{figure}[h]\label{figure}
\centering
\begin{tikzpicture}
\clip (-3.1,-3.1) rectangle (9,3.1);
\draw[thick,gray,dotted] (-3.5,0) -- (9,0);
\foreach \x in {0,15.65217,31.30435,...,350}
{
\draw[gray!50] (0,0) -- (\x:3);
}
\draw[black,thick] (0,0) circle (3cm);
\draw (0,0) node[circle,inner sep=2,fill = black,label=left:{$0$}] (0) {};
\draw (46.9563:3) node[circle,inner sep=2,fill = black,label=right:{$\omega^{m+1}$}] (m+1) {};
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=right:{$\omega^{-m}$}] (-m) {};
\draw[red,very thick] (31.3043:2.7) arc (31.3043:-31.3043:2.7);
\draw[] (2.5,-0.75) node[draw=none,label=above:{\textcolor{red}{$I$}}] () {};
\draw[red,thick] (31.3043:2.6) -- (31.3043:2.8);
\draw[red,thick] (-30.9:2.6) -- (-30.9:2.8);
\begin{scope}[shift=(m+1)]
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=above:{~~~~$\omega^{m+1}+\omega^{-m}$}] (pip) {};
\draw[gray!50] (0,0) -- (pip);
\varepsilonnd{scope}
\draw (46.9563:3) node[circle,inner sep=2,fill = black] (m+1) {};
\draw[black,thick] (0,0) -- (m+1);
\draw[black,thick] (0,0) -- (-m);
\draw[black,thick] (0,0) -- (pip);
\draw (0:4.5) node[circle,inner sep=2, fill=black,label=below:{$\fourier{\mathbbm{1}_{I^-}}(1)$}] (I-) {};
\draw[thick] (0,0) -- (I-);
\begin{scope}[shift=(I-)]
\draw[] (7.8261:4) node[circle,inner sep=2, fill=black,label=above:{$\fourier{\mathbbm{1}_{I'}}(1)$}] (pipshift) {};
\draw[gray!50] (0,0) -- (pipshift);
\varepsilonnd{scope}
\draw (0,0) -- (pipshift);
\draw[gray!50] (pipshift) -- (pip);
\draw[] (I-) node[circle,inner sep=2, fill=black] () {};
\varepsilonnd{tikzpicture}
\varepsilonnd{figure}
Suppose now that $a$ is even and let $m := (a-2)/2 \in [1,\frac{p-5}{4}]$.
Again without loss of generality we may redefine $I := [-m,m+1]$ and $I' := \lbrace -m-1\rbrace \cup [ -m+1,m+1]$.
Let also $I^- := [-m+1,m]$, which is non-empty.
The argument of $\fourier{\mathbbm{1}_{I^-}}(1)$ is $-\pi/p$.
Further, $\fourier{\mathbbm{1}_{I'}}(1) = \fourier{\mathbbm{1}_{I^-}}(1) + \omega^{m+1}+\omega^{-(m+1)}$.
The argument of $\omega^{m+1}+\omega^{-(m+1)}$ is $0$, so as before the argument of $\fourier{\mathbbm{1}_{I'}}(1)$ is strictly between $-\pi/p$ and $0$, as required.
\varepsilonpf
We say that an $a$-subset $A$ is a \varepsilonmph{punctured interval} if $A=I'+t$ or $A = -I'+t$ for some $t \in \Z p$.
That is, $A$ can be obtained from an interval of length $a+1$ by removing a penultimate point.
\begin{lemma}\label{int-equivalence}
Let $p \geqslantq 7$ be prime and let $a \in \lbrace 3,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},p-3\rbrace$.
Then the sets $I,I'\subseteq \Z p$ are not affine equivalent.
Thus no punctured interval is affine equivalent to an interval.
\varepsilonnd{lemma}
\bpf Suppose on the contrary that there is $\alpha \in \mathcal{A}$ with $\alpha(I')=I$. Let a \varepsilonmph{reflection} mean an affine map $R_c$ with $c\in\Z p$ that maps $x$ to $-x+c$.
Clearly, $I=[a]$ is invariant under the reflection $R:=R_{a-1}$. Thus $I'$ is invariant under the map $R':=\alpha^{-1}\circ R\circ \alpha$. As is easy to see, $R'$ is also some reflection and thus preserves the cyclic distances in $\Z p$. So $R'$ has to fix $a$, the unique element of $I'$ with both distance-1 neighbours lying outside of $I'$. Furthermore, $R'$ has to fix $a-2$, the unique element of $I'$ at distance 2 from $a$. However, no reflection can fix two distinct elements of $\Z p$, a contradiction.
\varepsilonpf
We remark that the previous lemma can also be deduced from Proposition~\ref{angleprop}. Indeed, for any $A \subseteq \Z p$, the multiset of Fourier coefficients of $A$ is the same as that of $x\cdot A$ for $x \in \Z p\setminus\lbrace 0 \rbrace$, and translating a subset changes the argument of Fourier coefficients by an integer multiple of $2\pi/p$. Thus for every subset which is affine equivalent to $I$, the argument of each of its Fourier coefficients is an integer multiple of $\pi/p$.
Let
$$
\rho(A) := \max_{\gamma \in \Z p \setminus \lbrace 0 \rbrace}|\fourier{\mathbbm{1}}_A(\gamma)|\quad\text{and}\quad R(a) := \leqslantft\lbrace \rho(A) : A \in \binom{\Z p}{a}\right\rbrace = \lbrace m_1(a) > m_2(a) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} \rbrace.
$$
Given $j \geqslantq 1$, we say that $A$ \varepsilonmph{attains} $m_j(a)$, and specifically that \varepsilonmph{$A$ attains $m_j(a)$ at $\gamma$} if $m_j(a) = \rho(A)=|\fourier{\mathbbm{1}_A}(\gamma)|$.
Notice that, since $\fourier{\mathbbm{1}_A}(-\gamma)=\overline{\fourier{\mathbbm{1}_A}(\gamma)}$, the set $A$ attains $m_j(a)$ at $\gamma$ if and only if $A$ attains $m_j(a)$ at $-\gamma$ (and $\gamma,-\gamma \neq 0$ are distinct values).
As we show in the next lemma, the $a$-subsets which attain $m_1(a)$ are precisely the affine images of $I$ (i.e.~arithmetic progressions), and the $a$-subsets which attain $m_2(a)$ are the affine images of the punctured interval~$I'$.
\begin{lemma}\label{lm:MaxNontriv} Let $p\geqslantq 7$ be prime and let $a\in [3,p-3]$. Then $|R(a)| \geqslantq 2$ and
\begin{itemize}
\item[(i)] $A \in \binom{\Z p}{a}$ attains $m_1(a)$ if and only if $A$ is affine equivalent to $I$, and every interval attains $m_1(a)$ at $1$ and $-1$ only;
\item[(ii)] $B\in\binom{\Z p}{a}$ attains $m_2(a)$ if and only if $B$ is affine equivalent to $I'$, and every punctured interval attains $m_2(a)$ at $1$ and $-1$ only.
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\bpf
Given $D \in \binom{\Z p}{a}$, we claim that there is some $D_{\rm{pri}} \in \binom{\Z p}{a}$ with the following properties:
\begin{itemize}
\item $D_{\rm{pri}}$ is affine equivalent to $D$;
\item $\rho(D) = |\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)|$; and
\item $-\pi/p < \arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) \leqslantq \pi/p$.
\varepsilonnd{itemize}
Call such a $D_{\rm{pri}}$ a \varepsilonmph{primary image} of $D$.
Indeed, suppose that $\rho(D) = |\fourier{\mathbbm{1}_D}(\gamma)|$ for some non-zero $\gamma \in \Z p$, and let $\fourier{\mathbbm{1}_D}(\gamma) = r'e^{\theta' i}$ for some $r' > 0$ and $0 \leqslantq \theta' < 2\pi$.
(Note that we have $r'>0$ since $p$ is prime.)
Choose $\varepsilonll \in \lbrace 0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},p-1\rbrace$ and $-\pi/p < \phi \leqslantq \pi/p$ such that $\theta' = 2\pi \varepsilonll/p + \phi$.
Let $D_{\rm{pri}} := \gamma\cdot D + \varepsilonll$. Then
$$
|\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)| = \leqslantft|\sum_{x \in D}\omega^{-\gamma x - \varepsilonll}\right| = |\omega^{-\varepsilonll} \fourier{\mathbbm{1}_D}(\gamma)| = |\fourier{\mathbbm{1}_D}(\gamma)| = \rho(D),
$$
and
$$
\arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) = \arg(e^{\theta' i}\omega^{-\varepsilonll}) = 2\pi \varepsilonll/p + \phi - 2\pi \varepsilonll/p = \phi,
$$
as required.
Let $D \subseteq \Z p$ have size $a$ and write $\fourier{\mathbbm{1}_D}(1) = re^{\theta i}$.
Assume by the above that $-\pi/p < \theta \leqslantq \pi/p$.
For all $j \in \Z p$, let
$$
h(j) := \Re(\omega^{-j}e^{-\theta i}) = \cos\leqslantft(\frac{2\pi j}{p}+\theta\right),
$$
where $\Re(z)$ denotes the real part of $z\in\mathbb{C}$.
Given any $a$-subset $E$ of $\Z p$, we have
\begin{equation}\label{hmbound}
H_D(E) := \sum_{j \in E}h(j) = \Re\leqslantft(e^{-\theta i}\sum_{j \in E}\omega^{-j}\right) = \Re\leqslantft(e^{-\theta i} \fourier{\mathbbm{1}_E}(1)\right) \leqslantq |\fourier{\mathbbm{1}_E}(1)|.
\varepsilonnd{equation}
Then
\begin{equation}\label{HAA}
H_D(D) = \sum_{j \in D}h(j) = \Re(e^{-\theta i} \fourier{\mathbbm{1}_D}(1)) = r = |\fourier{\mathbbm{1}_D}(1)|.
\varepsilonnd{equation}
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As we show in the next lemma, the $a$-subsets which attain $m_1(a)$ are precisely the affine images of $I$ (i.e.~arithmetic progressions), and the $a$-subsets which attain $m_2(a)$ are the affine images of the punctured interval~$I'$.
\begin{lemma}\label{lm:MaxNontriv} Let $p\geqslantq 7$ be prime and let $a\in [3,p-3]$. Then $|R(a)| \geqslantq 2$ and
\begin{itemize}
\item[(i)] $A \in \binom{\Z p}{a}$ attains $m_1(a)$ if and only if $A$ is affine equivalent to $I$, and every interval attains $m_1(a)$ at $1$ and $-1$ only;
\item[(ii)] $B\in\binom{\Z p}{a}$ attains $m_2(a)$ if and only if $B$ is affine equivalent to $I'$, and every punctured interval attains $m_2(a)$ at $1$ and $-1$ only.
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\bpf
Given $D \in \binom{\Z p}{a}$, we claim that there is some $D_{\rm{pri}} \in \binom{\Z p}{a}$ with the following properties:
\begin{itemize}
\item $D_{\rm{pri}}$ is affine equivalent to $D$;
\item $\rho(D) = |\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)|$; and
\item $-\pi/p < \arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) \leqslantq \pi/p$.
\varepsilonnd{itemize}
Call such a $D_{\rm{pri}}$ a \varepsilonmph{primary image} of $D$.
Indeed, suppose that $\rho(D) = |\fourier{\mathbbm{1}_D}(\gamma)|$ for some non-zero $\gamma \in \Z p$, and let $\fourier{\mathbbm{1}_D}(\gamma) = r'e^{\theta' i}$ for some $r' > 0$ and $0 \leqslantq \theta' < 2\pi$.
(Note that we have $r'>0$ since $p$ is prime.)
Choose $\varepsilonll \in \lbrace 0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},p-1\rbrace$ and $-\pi/p < \phi \leqslantq \pi/p$ such that $\theta' = 2\pi \varepsilonll/p + \phi$.
Let $D_{\rm{pri}} := \gamma\cdot D + \varepsilonll$. Then
$$
|\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)| = \leqslantft|\sum_{x \in D}\omega^{-\gamma x - \varepsilonll}\right| = |\omega^{-\varepsilonll} \fourier{\mathbbm{1}_D}(\gamma)| = |\fourier{\mathbbm{1}_D}(\gamma)| = \rho(D),
$$
and
$$
\arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) = \arg(e^{\theta' i}\omega^{-\varepsilonll}) = 2\pi \varepsilonll/p + \phi - 2\pi \varepsilonll/p = \phi,
$$
as required.
Let $D \subseteq \Z p$ have size $a$ and write $\fourier{\mathbbm{1}_D}(1) = re^{\theta i}$.
Assume by the above that $-\pi/p < \theta \leqslantq \pi/p$.
For all $j \in \Z p$, let
$$
h(j) := \Re(\omega^{-j}e^{-\theta i}) = \cos\leqslantft(\frac{2\pi j}{p}+\theta\right),
$$
where $\Re(z)$ denotes the real part of $z\in\mathbb{C}$.
Given any $a$-subset $E$ of $\Z p$, we have
\begin{equation}\label{hmbound}
H_D(E) := \sum_{j \in E}h(j) = \Re\leqslantft(e^{-\theta i}\sum_{j \in E}\omega^{-j}\right) = \Re\leqslantft(e^{-\theta i} \fourier{\mathbbm{1}_E}(1)\right) \leqslantq |\fourier{\mathbbm{1}_E}(1)|.
\varepsilonnd{equation}
Then
\begin{equation}\label{HAA}
H_D(D) = \sum_{j \in D}h(j) = \Re(e^{-\theta i} \fourier{\mathbbm{1}_D}(1)) = r = |\fourier{\mathbbm{1}_D}(1)|.
\varepsilonnd{equation}
Note that $H_D(E)$ is the (signed) length of the orthogonal projection of $\fourier{\mathbbm{1}_E}(1)\in\mathbb{C}$ on the 1-dimensional
line $\{xe^{i\theta}: x\in\I R\}$. As stated in~\varepsilonqref{hmbound} and~\varepsilonqref{HAA}, $H_D(E)\leqslant |\fourier{\mathbbm{1}_E}(1)|$
and this is equality for $E=D$. (Both of these facts are geometrically obvious.) If $|\fourier{\mathbbm{1}_D}(1)|=m_1(a)$ is maximum, then no
$H_D(E)$ for an $a$-set $E$ can exceed $m_1(a)=H_D(D)$. Informally speaking, the main idea of the proof is that if we fix the direction $e^{i\theta}$, then the projection length is maximised if we take $a$ distinct elements $j\in \I Z_p$ with the $a$ largest values of $h(j)$, that is, if we take some interval (with the runner-up being a punctured interval).
Let us provide a formal statement and proof of this now.
\begin{claim}\label{claim}
Let $\mathcal{I}_a$ be the set of length-$a$ intervals in $\Z p$.
\begin{itemize}
\item[(i)]
Let $M_1(D) \subseteq \binom{\Z p}{a}$ consist of $a$-sets $E\subseteq \Z p$ such that $H_D(E) \geqslantq H_D(C)$ for all $C \in \binom{\Z p}{a}$.
Then $M_1(D) \subseteq \mathcal{I}_a$.
\item[(ii)] Let $M_2(D) \subseteq \binom{\Z p}{a}$ be the set of $E \notin \mathcal{I}_a$ for which $H_D(E) \geqslantq H_D(C)$ for all $C \in \binom{\Z p}{a} \setminus \mathcal{I}_a$.
Then every $E \in M_2(A)$ is a punctured interval.
\varepsilonnd{itemize}
\varepsilonnd{claim}
\bpf
Suppose that $0 < \theta < \pi/p$.
Then $h(0) > h(1) > h(-1) > h(2) > h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} > h(\frac{p-1}{2}) > h(-\frac{p-1}{2})$.
In other words, $h(j_\varepsilonll) > h(j_k)$ if and only if $\varepsilonll < k$, where $j_m := (-1)^{m-1}\lceil m/2\rceil$.
Letting $J_{a-1} := \lbrace j_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},j_{a-2}\rbrace$, we see that
$$
H_D(J_{a-1} \cup \lbrace j_{a-1}\rbrace) > H_D(J_{a-1} \cup \lbrace j_{a}\rbrace) > H_D(J_{a-1} \cup \lbrace j_{a+1}\rbrace), H_D(J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace) > H_D(J)
$$
for all other $a$-subsets $J$.
But $J_{a-1} \cup \lbrace j_{a-1}\rbrace$ and $J_{a-1} \cup \lbrace j_a\rbrace$ are both intervals, and $J_{a-1} \cup \lbrace j_{a+1}\rbrace$ and $J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace$ are both punctured intervals.
So in this case $M_1(D) := \lbrace J_{a-1}\cup\lbrace j_{a-1}\rbrace\rbrace$ and $M_2(D) \subseteq \lbrace J_{a-1}\cup\lbrace j_{a+1}\rbrace, J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace\rbrace$, as required.
The case when $-\pi/p < \theta < 0$ is almost identical except now $j_\varepsilonll := (-1)^\varepsilonll\lceil \varepsilonll/2\rceil$ for all $0 \leqslantq \varepsilonll \leqslantq p-1$.
If $\theta=0$ then $h(0) > h(1) = h(-1) > h(2) = h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} > h(\frac{p-1}{2}) = h(-\frac{p-1}{2})$.
If $\theta=-\pi/p$ then $h(0)=h(-1) > h(1)=h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} = h(-\frac{p-1}{2}) > h(\frac{p-1}{2})$.
\varepsilonpf
{\mathrm e}dskip
\noindent
We can now prove part~(i) of the lemma.
Suppose $A \in \binom{\Z p}{a}$ attains $m_1(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Then the primary image $D$ of $A$ satisfies $|\fourier{\mathbbm{1}_D}(1)|=m_1(a)=|\fourier{\mathbbm{1}_A}(\gamma)|$.
So, for any $E \in M_1(D)$,
$$
|\fourier{\mathbbm{1}_A}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| \stackrel{(\ref{HAA})}{=} H_D(D) \leqslantq H_D(E) \stackrel{(\ref{hmbound})}{\leqslantq} |\fourier{\mathbbm{1}_E}(1)|,
$$
with equality in the first inequality if and only if $D \in M_1(D)$.
Thus, by Claim~\ref{claim}(i), $D$ is an interval, and so $A$ is affine equivalent to an interval, as required.
Further, if $A$ is an interval then $D$ is an interval if and only if $\gamma=\pm 1$.
This completes the proof of (i).
{\mathrm e}dskip
\noindent
For~(ii), note that $m_2(a)$ exists since by Lemma~\ref{int-equivalence}, there is a subset (namely $I'$) which is not affine equivalent to $I$. By~(i), it does not attain $m_1(a)$, so $\rho(I') \leqslantq m_2(a)$.
Suppose now that $B$ is an $a$-subset of $\Z p$ which attains $m_2(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Let $D$ be the primary image of $B$.
Then $D$ is not an interval.
This together with Claim~\ref{claim}(i) implies that $H_D(D) < H_D(E)$ for any $E \in M_1(D)$.
Thus, for any $C \in M_2(D)$, we have
$$
m_2(a) = |\fourier{\mathbbm{1}_B}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| = H_D(D) \leqslantq H_D(C) \leqslantq |\fourier{\mathbbm{1}_C}(1)|.
$$
with equality in the first inequality if and only if $D \in M_2(D)$.
Since $C$ is a punctured interval, it is not affine equivalent to an interval. So the first part of the lemma implies that $|\fourier{\mathbbm{1}_C}(1)|\leqslant m_2(a)$.
Thus we have equality everywhere and so $D \in M_2(D)$.
Therefore $B$ is the affine image of a punctured interval, as required.
Further, if $B$ is a punctured interval, then $D$ is a punctured interval if and only if $\gamma=\pm 1$.
This completes the proof of (ii).
\varepsilonpf
We will now prove Theorem~\ref{th:knot1}.
\bpf[Proof of Theorem~\ref{th:knot1}.]
Recall that $p\geqslant 7$, $a\in [3,p-3]$ and $k > k_0(a,p)$ is sufficiently large with $k\not\varepsilonquiv 1\pmod p$. Let $I = [a]$.
Given $t \in \Z p$, write $\rho_t := (\fourier{\mathbbm{1}_{I+t}}(1))^k\overline{\fourier{\mathbbm{1}_{I+t}}(1)}$ as $r_te^{\theta_t i}$, where $\theta_t \in [0,2\pi)$ and $r_t > 0$. Then~(\ref{skI2}) says that $\theta_t$ equals $-\pi(2t+a-1)(k-1)/p$ modulo $2\pi$.
Increasing $t$ by $1$ rotates $\rho_t$ by $-2\pi(k-1)/p$. Using the fact that $k-1$ is invertible modulo $p$, we have the following.
If $(a-1)(k-1)$ is even, then the set of $\theta_t$ for $t \in \Z p$ is precisely $0,2\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-2)\pi/p$, so there is a unique $t$ (resp.\ a unique $t'$) in $\Z p$ for which $\theta_t=\pi+\pi/p$ (resp.\ $\theta_{t'} = \pi-\pi/p$). Furthermore, $t' = -(a-1)-t$ and
$I+t' = -(I+t)$; thus $I+t$ and $I+t'$ have the same set of dilations.
If $(a-1)(k-1)$ is odd, then the set of $\theta_t$ for $t \in \Z p$ is precisely $\pi/p,3\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-1)\pi/p$, so there is a unique $t \in \Z p$ for which $\theta_t = \pi$.
We call $t$ (and $t'$, if it exists) \varepsilonmph{optimal}.
Let $t$ be optimal.
To prove the theorem, we will show that $F(\xi\cdot(I+t)) < F(A)$ (and so $s_k(\xi\cdot(I+t))<s_k(A)$) for any $a$-subset $A\subseteq\Z p$ which is not a dilation of $I+t$.
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{\mathrm e}dskip
\noindent
We can now prove part~(i) of the lemma.
Suppose $A \in \binom{\Z p}{a}$ attains $m_1(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Then the primary image $D$ of $A$ satisfies $|\fourier{\mathbbm{1}_D}(1)|=m_1(a)=|\fourier{\mathbbm{1}_A}(\gamma)|$.
So, for any $E \in M_1(D)$,
$$
|\fourier{\mathbbm{1}_A}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| \stackrel{(\ref{HAA})}{=} H_D(D) \leqslantq H_D(E) \stackrel{(\ref{hmbound})}{\leqslantq} |\fourier{\mathbbm{1}_E}(1)|,
$$
with equality in the first inequality if and only if $D \in M_1(D)$.
Thus, by Claim~\ref{claim}(i), $D$ is an interval, and so $A$ is affine equivalent to an interval, as required.
Further, if $A$ is an interval then $D$ is an interval if and only if $\gamma=\pm 1$.
This completes the proof of (i).
{\mathrm e}dskip
\noindent
For~(ii), note that $m_2(a)$ exists since by Lemma~\ref{int-equivalence}, there is a subset (namely $I'$) which is not affine equivalent to $I$. By~(i), it does not attain $m_1(a)$, so $\rho(I') \leqslantq m_2(a)$.
Suppose now that $B$ is an $a$-subset of $\Z p$ which attains $m_2(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Let $D$ be the primary image of $B$.
Then $D$ is not an interval.
This together with Claim~\ref{claim}(i) implies that $H_D(D) < H_D(E)$ for any $E \in M_1(D)$.
Thus, for any $C \in M_2(D)$, we have
$$
m_2(a) = |\fourier{\mathbbm{1}_B}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| = H_D(D) \leqslantq H_D(C) \leqslantq |\fourier{\mathbbm{1}_C}(1)|.
$$
with equality in the first inequality if and only if $D \in M_2(D)$.
Since $C$ is a punctured interval, it is not affine equivalent to an interval. So the first part of the lemma implies that $|\fourier{\mathbbm{1}_C}(1)|\leqslant m_2(a)$.
Thus we have equality everywhere and so $D \in M_2(D)$.
Therefore $B$ is the affine image of a punctured interval, as required.
Further, if $B$ is a punctured interval, then $D$ is a punctured interval if and only if $\gamma=\pm 1$.
This completes the proof of (ii).
\varepsilonpf
We will now prove Theorem~\ref{th:knot1}.
\bpf[Proof of Theorem~\ref{th:knot1}.]
Recall that $p\geqslant 7$, $a\in [3,p-3]$ and $k > k_0(a,p)$ is sufficiently large with $k\not\varepsilonquiv 1\pmod p$. Let $I = [a]$.
Given $t \in \Z p$, write $\rho_t := (\fourier{\mathbbm{1}_{I+t}}(1))^k\overline{\fourier{\mathbbm{1}_{I+t}}(1)}$ as $r_te^{\theta_t i}$, where $\theta_t \in [0,2\pi)$ and $r_t > 0$. Then~(\ref{skI2}) says that $\theta_t$ equals $-\pi(2t+a-1)(k-1)/p$ modulo $2\pi$.
Increasing $t$ by $1$ rotates $\rho_t$ by $-2\pi(k-1)/p$. Using the fact that $k-1$ is invertible modulo $p$, we have the following.
If $(a-1)(k-1)$ is even, then the set of $\theta_t$ for $t \in \Z p$ is precisely $0,2\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-2)\pi/p$, so there is a unique $t$ (resp.\ a unique $t'$) in $\Z p$ for which $\theta_t=\pi+\pi/p$ (resp.\ $\theta_{t'} = \pi-\pi/p$). Furthermore, $t' = -(a-1)-t$ and
$I+t' = -(I+t)$; thus $I+t$ and $I+t'$ have the same set of dilations.
If $(a-1)(k-1)$ is odd, then the set of $\theta_t$ for $t \in \Z p$ is precisely $\pi/p,3\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-1)\pi/p$, so there is a unique $t \in \Z p$ for which $\theta_t = \pi$.
We call $t$ (and $t'$, if it exists) \varepsilonmph{optimal}.
Let $t$ be optimal.
To prove the theorem, we will show that $F(\xi\cdot(I+t)) < F(A)$ (and so $s_k(\xi\cdot(I+t))<s_k(A)$) for any $a$-subset $A\subseteq\Z p$ which is not a dilation of $I+t$.
We will first show that $F(I+t)<F(A)$ for any $a$-subset $A$ which is not affine equivalent to an interval.
By Lemma~\ref{lm:MaxNontriv}(i), we have that $|\fourier{\mathbbm{1}_{I+t}}(\pm 1)|=m_1(a)$ and $\rho(A) \leqslantq m_2(a)$.
Let $m_2'(a)$ be the maximum of $\fourier{\mathbbm{1}_J}(\gamma)$ over all length-$a$ intervals $J$ and $\gamma \in [2,p-2]$.
Lemma~\ref{lm:MaxNontriv}(i) implies that $m_2'(a)<m_1(a)$.
Thus
\begin{eqnarray}\label{knot1eq2}
\leqslantft|F(I+t)-2(m_1(a))^{k+1}\cos(\theta_t) - F(A)\right| \leqslantq (p-1)(m_2(a))^{k+1} + (p-3)\leqslantft(m_2'(a)\right)^{k+1}.
\varepsilonnd{eqnarray}
Now $\cos(\theta_t) \leqslantq \cos(\pi-\pi/p) < -0.9$ since $p \geqslantq 7$.
This together with the fact that $k\geqslant k_0(a,p)$ and Lemma~\ref{lm:MaxNontriv} imply that the absolute value of $2(m_1(a))^{k+1}\cos(\theta_t)<0$ is greater than the right-hand size of~(\ref{knot1eq2}).
Thus $F(I+t) < F(A)$, as required.
The remaining case is when $A=\zeta\cdot(I+v)$ for some non-optimal $v \in \Z p$ and non-zero $\zeta \in \Z p$. Since $s_k(A)=s_k(I+v)$, we may assume that $\zeta=1$.
Note that $\cos(\theta_t) \leqslantq \cos(\pi-\pi/p) < \cos(\pi-2\pi/p) \leqslantq \cos(\theta_v)$.
Thus
\begin{align*}
F(I+t)-F(I+v) &\leqslantq 2(m_1(a))^{k+1}(\cos(\theta_t)-\cos(\theta_v)) + (2p-4)(m_2'(a))^{k+1}\\
&\leqslantq 2(m_1(a))^{k+1}(\cos(\pi-\pi/p)-\cos(\pi-2\pi/p)) + (2p-4)(m_2'(a))^{k+1} <0
\varepsilonnd{align*}
where the last inequality uses the fact that $k$ is sufficiently large.
Thus $F(I+t)<F(I+v)$, as required.
\varepsilonpf
Finally, using similar techniques, we prove Theorem~\ref{th:k1}.
\bpf[Proof of Theorem~\ref{th:k1}.]
Recall that $p\geqslant 7$, $a\in [3,p-3]$ and $k > k_0(a,p)$ is sufficiently large with $k\varepsilonquiv1\pmod p$. Let $I:=[a]$ and $I'=[a-1]\cup\{a\}$.
Suppose first that $a$ and $k$ are both even. Let $A\subseteq\Z p$ be
an arbitrary $a$-set not affine equivalent to the interval~$I$.
By Lemma~\ref{lm:MaxNontriv}, $I$ attains $m_1(a)$ (exactly at $x=\pm 1$), while $\rho(A)<m_1(a)$. Also, $m_2'(a)<m_1(a)$, where $
m_2'(a):=\max_{\gamma\in [2,p-2]}|\fourier{\mathbbm{1}_I}(\gamma)|$. Thus
\begin{eqnarray*}
F(I) - F(A) &\stackrel{(\ref{eq:SkF1}),(\ref{skI})}{\leqslantq}& 2\sum_{\gamma=1}^{\frac{p-1}{2}} (-1)^\gamma\leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1} + \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_A}(\gamma)\right|^{k+1}\\
&\leqslantq& -2(m_1(a))^{k+1} + (2p-4) (\max\{m_2(a),m_2'(a)\})^{k+1}\ <\ 0,
\varepsilonnd{eqnarray*}
where the last inequality uses the fact that $k$ is sufficiently large.
So $s_k(a)=s_k(I)$.
Using Lemma~\ref{lm:MaxNontriv}, the same argument shows that, for all $B \in \binom{\Z p}{a}$, we have $s_k(B)=s_k(a)$ if and only if $B$ is an affine image of $I$.
This completes the proof of Part 1 of the theorem.
Suppose now that at least one of $a,k$ is odd. Let $A$ be an $a$-set not equivalent to~$I$. Again by Lemma~\ref{lm:MaxNontriv}, we have
\begin{eqnarray*}
F(I) - F(A) &\geqslantq& \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1} - \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_A}(\gamma)\right|^{k+1}\\
&\geqslantq& 2(m_1(a))^{k+1} - (p-1)(m_2(a))^{k+1} \ >\ 0.
\varepsilonnd{eqnarray*}
So the interval $I$ and its affine images have in fact the largest number of additive $(k+1)$-tuples among all $a$-subsets of $\Z p$. In particular, $s_k(a) < s_k(I)$.
Suppose that there is some $A \in \binom{\Z p}{a}$ which is not affine equivalent to $I$ or~$I'$. (If there is no such $A$, then the unique extremal sets are affine images of $I'$ for all $k > k_0(a,p)$, giving the required.)
Write $\rho := re^{\theta i} = \fourier{\mathbbm{1}_{I'}}(1)$.
Then by Lemma~\ref{lm:MaxNontriv}(ii), we have $r=m_2(a)$, and $\rho(A) \leqslantq m_3(a)$.
Given $k \geqslantq 2$, let $s \in \mathbb{N}$ be such that $k=sp+1$. Then
\begin{equation}\label{FI'}
\Big|F(I') - 2m_2(a)^{k+1}\cos (sp\theta)-F(A)\Big|\leqslant (p-1)m_3(a)^{k+1}+(p-3)\leqslantft(m_2'(a)\right)^{k+1}.
\varepsilonnd{equation}
Proposition~\ref{angleprop} implies that there is an even integer $\varepsilonll \in \I N$
for which $c := p\theta - \varepsilonll\pi \in (-\pi,\pi)\setminus\{0\}$. Let $\varepsilon := \frac{1}{3}\min\lbrace |c|,\pi-|c|\rbrace > 0$.
Given an integer $t$, say that $s\in\I N$ is \varepsilonmph{$t$-good} if $sc \in ((t-\frac{1}{2})\pi+\varepsilon,(t+\frac{1}{2})\pi-\varepsilon)$.
This real interval has length $\pi-2\varepsilon > |c|>0$, so must contain at least one integer multiple of $c$.
In other words, for all $t \in \mathbb{Z}\setminus\{0\}$ with the same sign as $c$, there exists a $t$-good integer $s> 0$.
As $sp\theta\varepsilonquiv sc\pmod{2\pi}$, the sign of $\cos(sp\theta)$ is $(-1)^{t}$. Moreover, Lemma~\ref{lm:MaxNontriv} implies that
$m_2(a)> m_3(a), m'_2(a)$.
Thus, when $k=sp+1>k_0(a,p)$, the absolute value of $2m_2(a)^{k+1}\cos(sp\theta)$ is greater than the right-hand side of~(\ref{FI'}).
Thus, for large $|t|$, we have $F(A) > F(I')$ if $t$ is even and $F(A)<F(I')$ if $t$ is odd, implying the theorem by~\varepsilonqref{eq:SkF1}.
\varepsilonpf
\begin{thebibliography}{1}
\bibitem{Bajnok18acmrp}
B.~Bajnok, \varepsilonmph{Additive combinatorics: A menu of research problems}, CRC
Press, Roca Baton, FL, 2018.
\bibitem{Erdos65}
P.~Erd{\H o}s, \varepsilonmph{Extremal problems in number theory}, Proc. {S}ympos.
{P}ure {M}ath., {V}ol. {VIII}, Amer. Math. Soc., Providence, R.I., 1965,
pp.~181--189.
\bibitem{GreenRuzsa05}
B.~Green and I.~Z. Ruzsa, \varepsilonmph{Sum-free sets in abelian groups}, Israel J.\
Math. \textbf{147} (2005), 157--188.
\bibitem{HuczynskaMullenYucas09jcta}
S.~Huczynska, G.~L. Mullen, and J.~L. Yucas, \varepsilonmph{The extent to which subsets
are additively closed}, J. Combin. Theory Ser. A \textbf{116} (2009),
831--843.
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Suppose now that at least one of $a,k$ is odd. Let $A$ be an $a$-set not equivalent to~$I$. Again by Lemma~\ref{lm:MaxNontriv}, we have
\begin{eqnarray*}
F(I) - F(A) &\geqslantq& \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1} - \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_A}(\gamma)\right|^{k+1}\\
&\geqslantq& 2(m_1(a))^{k+1} - (p-1)(m_2(a))^{k+1} \ >\ 0.
\varepsilonnd{eqnarray*}
So the interval $I$ and its affine images have in fact the largest number of additive $(k+1)$-tuples among all $a$-subsets of $\Z p$. In particular, $s_k(a) < s_k(I)$.
Suppose that there is some $A \in \binom{\Z p}{a}$ which is not affine equivalent to $I$ or~$I'$. (If there is no such $A$, then the unique extremal sets are affine images of $I'$ for all $k > k_0(a,p)$, giving the required.)
Write $\rho := re^{\theta i} = \fourier{\mathbbm{1}_{I'}}(1)$.
Then by Lemma~\ref{lm:MaxNontriv}(ii), we have $r=m_2(a)$, and $\rho(A) \leqslantq m_3(a)$.
Given $k \geqslantq 2$, let $s \in \mathbb{N}$ be such that $k=sp+1$. Then
\begin{equation}\label{FI'}
\Big|F(I') - 2m_2(a)^{k+1}\cos (sp\theta)-F(A)\Big|\leqslant (p-1)m_3(a)^{k+1}+(p-3)\leqslantft(m_2'(a)\right)^{k+1}.
\varepsilonnd{equation}
Proposition~\ref{angleprop} implies that there is an even integer $\varepsilonll \in \I N$
for which $c := p\theta - \varepsilonll\pi \in (-\pi,\pi)\setminus\{0\}$. Let $\varepsilon := \frac{1}{3}\min\lbrace |c|,\pi-|c|\rbrace > 0$.
Given an integer $t$, say that $s\in\I N$ is \varepsilonmph{$t$-good} if $sc \in ((t-\frac{1}{2})\pi+\varepsilon,(t+\frac{1}{2})\pi-\varepsilon)$.
This real interval has length $\pi-2\varepsilon > |c|>0$, so must contain at least one integer multiple of $c$.
In other words, for all $t \in \mathbb{Z}\setminus\{0\}$ with the same sign as $c$, there exists a $t$-good integer $s> 0$.
As $sp\theta\varepsilonquiv sc\pmod{2\pi}$, the sign of $\cos(sp\theta)$ is $(-1)^{t}$. Moreover, Lemma~\ref{lm:MaxNontriv} implies that
$m_2(a)> m_3(a), m'_2(a)$.
Thus, when $k=sp+1>k_0(a,p)$, the absolute value of $2m_2(a)^{k+1}\cos(sp\theta)$ is greater than the right-hand side of~(\ref{FI'}).
Thus, for large $|t|$, we have $F(A) > F(I')$ if $t$ is even and $F(A)<F(I')$ if $t$ is odd, implying the theorem by~\varepsilonqref{eq:SkF1}.
\varepsilonpf
\begin{thebibliography}{1}
\bibitem{Bajnok18acmrp}
B.~Bajnok, \varepsilonmph{Additive combinatorics: A menu of research problems}, CRC
Press, Roca Baton, FL, 2018.
\bibitem{Erdos65}
P.~Erd{\H o}s, \varepsilonmph{Extremal problems in number theory}, Proc. {S}ympos.
{P}ure {M}ath., {V}ol. {VIII}, Amer. Math. Soc., Providence, R.I., 1965,
pp.~181--189.
\bibitem{GreenRuzsa05}
B.~Green and I.~Z. Ruzsa, \varepsilonmph{Sum-free sets in abelian groups}, Israel J.\
Math. \textbf{147} (2005), 157--188.
\bibitem{HuczynskaMullenYucas09jcta}
S.~Huczynska, G.~L. Mullen, and J.~L. Yucas, \varepsilonmph{The extent to which subsets
are additively closed}, J. Combin. Theory Ser. A \textbf{116} (2009),
831--843.
\bibitem{Lev01duke}
V.~F.~Lev, \varepsilonmph{Linear equations over $\mathbb{Z} / p\mathbb{Z}$ and moments of exponential sums}, Duke Math. J. \textbf{107} (2) (2001), 239--263.
\bibitem{NazarewiczObrienOneillStaples07}
E.~Nazarewicz, M.~O'Brien, M.~O'Neill, and C.~Staples, \varepsilonmph{Equality in
{P}ollard's theorem on set addition of congruence classes}, Acta Arith.
\textbf{127} (2007), 1--15.
\bibitem{Pollard75}
J.~M. Pollard, \varepsilonmph{Addition properties of residue classes}, J.\ Lond.\ Math.\
Soc. \textbf{11} (1975), 147--152.
\bibitem{SamotijSudakov16pmcps}
W.~Samotij and B.~Sudakov, \varepsilonmph{The number of additive triples in subsets of
{A}belian groups}, Math.\ Proc.\ Camb.\ Phil.\ Soc. \textbf{160} (2016),
495--512.
\bibitem{Terras99faofg}
Audrey Terras, \varepsilonmph{Fourier analysis on finite groups and applications},
London Mathematical Society Student Texts, vol.~43, Cambridge University
Press, Cambridge, 1999.
\varepsilonnd{thebibliography}
\varepsilonnd{document}
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\begin{document}
\title{Duality results for a general trigonometric approximation problem}
\begin{abstract}
Let \(\alpha\in\iva\) and \(\mu\) be a regular finite Borel measure on a locally compact abelian group.
The paper deals with a general trigonometric approximation problem in \(\Lam\), which arises in prediction theory of harmonizable symmetric \tas{} processes.
To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors.
The novelty of the present paper is that we do not make any additional assumption on \(\mu\).
Moreover, for \(\alpha=2\), multivariate extensions are obtained.
\end{abstract}
\begin{description}
\item[Keywords:] Regular Borel measure, space of \tai{} functions, trigonometric approximation, duality.
\end{description}
\section{Introduction}
Let \(w\) be a weight function on \((-\pi,\pi]\), \(S\) a nonvoid subset of the set \(\Z\) of integers, \(s\in S\), and \(\TZS\) the linear space of trigonometric polynomials with frequencies from \(\ZS\).
An important task in prediction theory of weakly stationary or, more generally, harmonizable symmetric \tas{} sequences is to compute the prediction error, \tie{}, the distance of the function \(\ec^{\iu s\cdot}\) to the set \(\TZS\) with respect to the metric of the Banach space \(\Law\), \(\alpha\in\iva\).
In 1984 Nakazi~\zita{MR766702} introduced a new idea into the study of this problem.
Among other things, his method opened a way for him to give an elegant proof of the celebrated Szeg\H{o} infimum formula.
Miamee and Pourahmadi~\zita{MR949088} pointed out that the essence of Nakazi's technique is a certain duality between the spaces \(\Law\) and \(\Ldaw\), \(\da \defeq\alpha/(\alpha-1)\), presumed that \(w^\inv\) exists and is integrable.
In the sequel this duality relation turned out to be rather fruitful.
It was applied to a variety of sets \(S\), modified, and extended to more general prediction problems, \tcf{}\ the papers~\zitas{MR1443377,MR1230571,MR1289507,MR2262930}, as well as~\zita{MR1170451} for fields on \(\Z^2\),~\zita{MR772194} for processes on discrete abelian groups,~\zita{MR1928917} for multivariate sequences.
Urbanik~\zita{MR1808671} defined a notion of a dual stationary sequence, \tcf{}~\zitaa{MR1849562}{\csec{8.5}} and~\zita{MR1899439}.
Kasahara, Pourahmadi and Inoue~\zita{MR2547424} used a modified duality method to obtain series representations of the predictor.
It should also be mentioned that \cthm{24} of~\zita{MR0009098} was, perhaps, the first published duality result of the type in question, \tcf{} its extension by Yaglom~\zitaa{MR0031214}{\cthm{2}}.
Many of the preceding results were obtained under the rather strong additional condition that \(w^\inv\) exists and is integrable, although for some special sets authors succeeded in weakening this condition, \tcf{}~\zitas{MR1443377,MR772194,MR2262930}.
In our paper we study the above described problem for spaces \(\Lam\), where the Borel measure \(\mu\) is not assumed to be absolutely continuous.
Moreover, motivated by Weron's paper~\zita{MR772194} we shall be concerned with regular finite Borel measures on locally compact abelian groups.
Since in the literature there exists differing definitions of a regular measure, \rsec{3} deals with the definition and some basic facts of regular measures.
The results of \rsecss{2}{4} show that under a condition, which is satisfied by many sets \(S\) occurring in application, one can assume that the measure \(\mu\) is absolutely continuous.
Establishing this result, we introduce a class of sets, which we call class of \tACs{s} and which, as it seems to us, deserves further investigation.
\rsec{5} gives a solution to the problem if \(\mu\) is absolutely continuous.
Unlike most of the authors above we do not make any condition on the corresponding Radon-Nikodym derivative.
In~\zita{MR1462266} there were defined Banach spaces of matrix-valued functions \tai{} with respect to a positive semidefinite matrix-valued measure, see \rsec{2} for the definition and basic facts.
Since part of our results can be easily generalized, we state and prove them in this more general framework.
If \(S\) is a singleton, the corresponding results can be used to obtain minimality criteria for multivariate stationary sequences.
We shall not go into detail but refer to the recent paper~\zita{KM15}, where various minimality notions were discussed.
If \(X\) is a matrix, denote by \(X^\ad\), \(X^\mpi\), and \(\ran{X}\) its adjoint, Moore-Penrose inverse, and range, \tresp{} The symbol \(\Oe\) stands for the zero element in an arbitrary linear space.
\section{The space $\LaM$}\label{2}
Let \(q\in\N\), \(\Mggq\) be the cone of all \tpsd{} (hence, \tH{}) \tqqa{matrices} with complex entries.
Let \(\OA\) be a measurable space and \(M\) an \taval{\Mggq} measure on \(\A\).
If \(\tau\) is a \tsf{} measure on \(\A\) such that \(M\) is absolutely continuous with respect to \(\tau\), denote by \(\dif M/\dif\tau\) the corresponding Radon-Nikodym derivative and by \(P(\omega)\) the orthoprojection in \(\Cq\) onto \(\ran{\rk{\dif M/\dif\tau}(\omega)}\), \(\omega\in\Omega\).
Let \(\no{\cdot}\) be the euclidean norm on \(\Cq\) and write the vectors of \(\Cq\) as column vectors.
Two \tAm{} \taval{\Cq} functions \(f\) and \(g\) are called \tMe{} if \(Pf=Pg\) \tae{\tau} For \(\alpha\in\iva\), denote by \(\LaM\) the space of all (\tMec{es} of) \tAm{} functions \(f\) such that \(\noa{f}\defeq\ek{\int_\Omega\no{\rk{\dif M/\dif\tau}^{1/\alpha}f}^\alpha\dif\tau}^{1/\alpha}<\infty\).
Recall that the definition of \(\LaM\) does not depend on the choice of \(\tau\) and that \(\LaM\) is a Banach space with respect to the norm \(\noa{\cdot}\).
Note that \(\LhM\) is a Hilbert space with inner product \(\int_\Omega g^\ad\rk{\dif M/\dif\tau}f\dif\tau\), \(f,g\in\LhM\), and that for \(q=1\), the space \(\LaM\) is the well known space of (equivalence classes of) \tAm{} \taval{\C} functions \tai{} with respect to \(M\).
The space \(\LhM\) was introduced by I.~S.~Kats~\zita{MR0080280} and in a somewhat more general form by Rosenberg~\zita{MR0163346}.
Both notions were applied in the theory of weakly stationary processes with the same success, \tcf{}~\zitas{MR0159363,MR0279952} for an application of Kats' and Rosenberg's definitions, \tresp{} An extension to \(\alpha\neq2\) was given by Duran and Lopez-Rodriguez~\zita{MR1462266}, \tcf{}~\zita{MR1160966} for a more general setting of operator-valued measures.
To simplify the presentation slightly we shall be concerned with the space \(\LaM\) as defined above, which can be considered as a generalization of Kats' definition to the case \(\alpha\neq2\).
\begin{lem}[\tcf{}~\zitaa{MR1462266}{\cthm{2.5}},~\zitaa{MR1160966}{\cthm{9}}]\label{L2.1}
Let \(\alpha\in\iva\) and \(\da \defeq\alpha/\rk{\alpha-1}\).
If \(\ell\) is a bounded linear functional on \(\LaM\), then there exists \(g\in\LdaM\) such that \(\ell(f)=\int_\Omega g^\ad\rk{\dif M/\dif\tau}f\dif\tau\) for all \(f\in\LaM\).
The correspondence \(\ell\mapsto g\) establishes an isometric isomorphism between the dual space of \(\LaM\) and the space \(\LdaM\).
\end{lem}
Let \(M_1\) and \(M_2\) be \taval{\Mggq} measures on \(\A\) such that \(M=M_1+M_2\) and \(M_1(A)=M_2(\Omega\setminus A)=\Oe\) for some set \(A\in\A\).
Let \(\ind{C}\) denote the indicator function of a set \(C\).
Identifying \(\LaMa\) with the space \(\indOA \LaM=\setaa{\indOA f}{f\in\LaM}\) and \(\LaMb\) with \(\indA \LaM\), one obtains a direct sum decomposition \(\LaM=\LaMa\dotplus\LaMb\).
For a linear subset \(\Lc\) of \(\LaM\), denote by \(\Lca\) its closure in \(\LaM\)
\begin{lem}\label{L2.2}
If
\begin{equation}\label{e2.1}
\LaMb
\subseteq\Lca,
\end{equation}
then
\begin{equation}\label{e2.2}
\cl{\rk{\indA \Lc}}
=\LaMb
\end{equation}
and
\begin{equation}\label{e2.3}
\Lca
=\cl{\rk{\indOA \Lc}}\dotplus\cl{\rk{\indA \Lc}}
=\cl{\rk{\indOA \Lc}}\dotplus\LaMb.
\end{equation}
\end{lem}
\begin{proof}
The continuity of the map \(f\mapsto\indA f\), \(f\in\LaM\), and condition \eqref{e2.1} yield \(\LaMb=\indA\LaM\subseteq\indA\Lca\subseteq\cl{\rk{\indA\Lc}}\).
Since the inclusion \(\cl{\rk{\indA\Lc}}\subseteq\LaMb\) is obvious, equality \eqref{e2.2} follows.
For the proof of \eqref{e2.3}, note first that \(\Lca\subseteq\cl{\rk{\indOA\Lc}}\dotplus\cl{\rk{\indA\Lc}}\).
To prove the opposite inclusion, let \(f\in\Lca\).
By \eqref{e2.2} and \eqref{e2.1}, we have \(\indA f\in\indA\Lca\subseteq\cl{\rk{\indA\Lc}}=\LaMb\subseteq\Lca\), which gives \(\cl{\rk{\indA\Lc}}\subseteq\cl{\rk{\indA\Lca}}\subseteq\Lca\).
The relation \(\indOA f=f-\indA f\in\Lca\) implies that \(\cl{\rk{\indOA\Lc}}\subseteq\Lca\), hence, \(\cl{\rk{\indOA \Lc}}\dotplus\cl{\rk{\indA \Lc}}\subseteq\Lca\).
\end{proof}
\begin{lem}\label{L2.3}
Let \(f\in\LaM\) and \(\rho\defeq\inf\setaa{\noa{f-g}}{g\in\Lc}\), \(\rho_1\defeq\inf\setaa{\noa{\indOA(f-g)}}{g\in\Lc}\).
If \eqref{e2.1} is satisfied, then \(\rho=\rho_1\).
\end{lem}
\begin{proof}
For \(\epsilon>0\), there exist \(g_1,g_2\in\Lc\) such that \(\noa{\indOA(f-g_1)}<\rho_1+\epsilon\) and \(\noa{\indA(f-g_2)}<\epsilon\) according to \eqref{e2.2}.
By \rlem{L2.2}, \(\indOA g_1+\indA g_2\in\Lca\).
Therefore, \(\rho\leq\noa{f-\rk{\indOA g_1+\indA g_2}}\leq\noa{\indOA(f-g_1)}+\noa{\indA(f-g_2)}<\rho_1+2\epsilon\), hence, \(\rho\leq\rho_1\).
Since the inequality \(\rho_1\leq\rho\) is trivial, the result follows.
\end{proof}
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\section{Regular Borel measures on locally compact abelian groups}\label{3}
Let \(\Gamma\) be a locally compact abelian group, \(\BsaG\) the \tsa{} of Borel sets of \(\Gamma\), and \(\lambda\) a Haar measure on \(\BsaG\).
We recall the definition and some elementary properties of regular measures, however, we mention that we do not know an example of a non-regular finite Borel measure on a locally compact abelian group.
Therefore, some of the following assertions might be redundant.
A finite non-negative measure \(\mu\) on \(\BsaG\) is called
\begin{itemize}
\item[a)] \emph{outer regular} if \(\mu(B)=\inf\setaa{\mu(U)}{U\text{ is open and }B\subseteq U\subseteq\Gamma}\) for all \(B\in\BsaG\),
\item[b)] \emph{inner regular} if \(\mu(B)=\sup\setaa{\mu(K)}{K\text{ is compact and }K\subseteq B}\) for all \(B\in\BsaG\),
\item[c)] \emph{regular} if it is outer regular and inner regular,
\end{itemize}
\tcf{}~\zitaa{MR578344}{\cpage{206}} and~\zitaa{MR0156915}{(11.34)}.
\begin{lem}\label{L3.1}
\begin{aeqi}{0}
\item\label{L3.1.i} Let \(\mu\) be a finite non-negative measure on \(\BsaG\).
If it is inner regular, then it is outer regular.
\item\label{L3.1.ii} Any positive linear combination of regular finite non-negative measures is regular.
\item\label{L3.1.iii} Let \(\mu\) and \(\nu\) be finite non-negative measures on \(\BsaG\).
If \(\mu\) is regular and \(\nu\) is absolutely continuous with respect to \(\mu\), then \(\nu\) is regular.
\end{aeqi}
\end{lem}
\begin{proof}
To prove~\ref{L3.1.i} let \(B\in\BsaG\) and \(\co{B}\defeq\Gamma\setminus B\).
If \(\mu\) is inner regular, for \(\epsilon>0\) there exists a compact set \(K\subseteq\co{B}\) such that \(\mu(\co{B}\setminus K)<\epsilon\).
Since \(\Gamma\) is a Hausdorff space, the compact set \(K\) is closed, hence, the set \(\co{K}\) is open, and it satisfies \(B\subseteq\co{K}\) and \(\mu(\co{K}\setminus B)=\mu(\co{B}\setminus K)<\epsilon\), which implies that \(\mu\) is outer regular.
Assertion~\ref{L3.1.ii} is clear and assertion~\ref{L3.1.iii} is an immediate consequence of the following fact.
If \(\nu\) is absolutely continuous with respect to \(\mu\), for \(\epsilon>0\) there exists \(\delta>0\) such that for all \(B\in\BsaG\), the inequality \(\mu(B)<\delta\) yields \(\nu(B)<\epsilon\), \tcf{}~\zitaa{MR578344}{\clem{4.2.1}}.
\end{proof}
A \taval{\C} measure \(\mu\) on \(\BsaG\) is called \emph{regular} if its variation \(\var{\mu}\) is regular.
It is called \emph{absolutely continuous} or \emph{singular}, \tresp{}, if its variation is absolutely continuous or singular with respect to \(\lambda\).
\begin{lem}\label{L3.2}
Let \(\mu\) be an absolutely continuous and regular \taval{\C} measure on \(\BsaG\).
Then there exists a Radon-Nikodym derivative of \(\mu\) with respect to \(\lambda\).
\end{lem}
\begin{proof}
Since \(\mu\) is assumed to be regular, there exists a sequence \(\gk{K_n}_{n\in\N}\) of compact sets such that \(\var{\mu}(\Gamma\setminus K_n)<1/n\), \(n\in\N\).
If \(B\defeq\bigcup_{n=1}^\infty K_n\), then \(B\in\BsaG\), \(\var{\mu}(\Gamma\setminus B)=0\), and from \(\lambda(K_n)<\infty\), \(n\in\N\), we get that \(B\) is a set of \tsf{} \tam{\lambda}.
Thus, the assertion follows from the Radon-Nikodym theorem, \tcf{}~\zitaa{MR578344}{\cthm{4.2.3}}.
\end{proof}
By definition, an \taval{\Mggq} measure \(M\) on \(\BsaG\) is \emph{regular} if all its entries are regular.
Let \(\mu_{jk}\) be the measure at place \((j,k)\).
Since for all \(B\in\BsaG\), \(\abs{\mu_{jk}(B)}\leq\ek{\mu_{jj}(B)+\mu_{kk}(B)}/2\), hence \(\var{\mu_{jk}}(B)\leq\ek{\mu_{jj}(B)+\mu_{kk}(B)}/2\), from \rlem{L3.1} one can conclude that \(M\) is regular if and only if each measure on the principal diagonal is regular.
Taking into account \rlemp{L3.1}{L3.1.iii} and \rlem{L3.2}, we arrive at the following result.
\begin{lem}\label{L3.3}
If \(\ac{M}\) is the absolutely continuous part of a regular \taval{\Mggq} measure \(M\) on \(\BsaG\), then there exists the Radon-Nikodym derivative \(\dif\ac{M}/\dif\lambda\eqdef W\) of \(\ac{M}\) with respect to \(\lambda\).
\end{lem}
\section{A general trigonometric approximation problem in $\LaM$}\label{4}
For \(k\in\mn{1}{q}\) denote by \(\eu{k}\) the \tth{k} vector of the standard orthonormal basis of \(\Cq\).
Let \(G\) be a locally compact abelian group and \(\Gamma\) its dual.
The value of \(\gamma\in\Gamma\) at \(x\in G\) is denoted by \(\inner{\gamma}{x}\).
For \(x\in G\), define a function \(\chu{x}\) by \(\chua{x}{\gamma}\defeq\inner{\gamma}{x}\), \(\gamma\in\Gamma\), and set \(\chuu{x}{k}\defeq\chu{x}\eu{k}\).
Let \(S\) be a nonvoid subset of \(G\).
If \(\GsS\) is not empty, denote by \(\T{\GsS}\) the linear space of all \taval{\Cq} trigonometric polynomials on \(\Gamma\) with frequencies from \(\GsS\), \tie{}, \(t\in\T{\GsS}\) if and only if it has the form \(t=\sum_{j=1}^n\chu{x_j}u_j\), \(x_j\in\GsS\), \(u_j\in\Cq\), \(j\in\mn{1}{n}\), \(n\in\N\).
If \(\GsS\) is empty, let \(\T{\GsS}\) be the space consisting of the zero function on \(\Gamma\).
Let \(M\) be a regular \taval{\Mggq} measure on \(\BsaG\).
Motivated by problems of prediction theory we are interested in computing the distance
\begin{align*}
d&\defeq\inf\setaa*{\noa*{\chuu{s}{k}-t}}{t\in\T{\GsS}},&s&\in S,&k&\in\mn{1}{q}.
\end{align*}
\begin{thm}\label{T4.1}
Let \(M=\ac{M}+\si{M}\) be the Lebesgue decomposition of \(M\) into its absolutely continuous part \(\ac{M}\) and singular part \(\si{M}\).
If
\begin{equation}\label{T4.1.1}
\Loa{\alpha}{\si{M}}
\subseteq\cl{\ek*{\T{\GsS}}},
\end{equation}
then for \(s\in S\), \(k\in\mn{1}{q}\),
\[
d
=\inf\setaa*{\ek*{\int\no*{\rk*{\frac{\dif\ac{M}}{\dif\lambda}}^{1/\alpha}\rk{\chuu{s}{k}-t}}^\alpha\dif\lambda}^{1/\alpha}}{t\in\T{\GsS}},
\]
where here and in what follows the domain of integration is \(\Gamma\) if it is not indicated explicitly.
\end{thm}
The preceding theorem immediately follows from \rlemss{L2.3}{L3.3} and it shows that under condition \eqref{T4.1.1} it is enough to solve the approximation problem for the absolutely continuous part of \(M\).
The next section deals with its partial solution and the rest of the present section is devoted to subsets of \(G\) with the property that \eqref{T4.1.1} is satisfied for each regular \taval{\Mggq} measure.
For a regular \taval{\Cq} measure \(\mu\) on \(\BsaG\), denote by \(\fsinv{\mu}\) its inverse Fourier-Stieltjes transform, \tie{}\ \(\fsinv{\mu}(x)\defeq\int\inner{\gamma}{x}\mu(\dif\gamma)=\int\chu{x}\dif\mu\), \(x\in G\).
Let \(\LlCq\) be the Banach space of \tam{\BsaG}able \taval{\Cq} functions integrable with respect to \(\lambda\).
If \(f\in\LlCq\), the symbol \(\fsinv{f}\) stands for its inverse Fourier transform.
\begin{defn}\label{D4.2}
A subset \(S\) of \(G\) is called an \emph{\tACs{}}, if for all regular \taval{\Cq} measures \(\mu\) on \(\BsaG\), the equality \(\fsinv{\mu}(x)=0\) for all \(x\in S\) implies that \(\mu\) is absolutely continuous.
\end{defn}
\begin{lem}\label{L4.3}
If \(\GsS\) is an \tACs{}, then \eqref{T4.1.1} is satisfied.
\end{lem}
\begin{proof}
Assume that there exists \(f\in\Loa{\alpha}{\si{M}}\), which does not belong to \(\cl{\ek{\T{\GsS}}}\).
Then by \rlem{L2.1} there exists \(g\in\LdaM\) such that
\begin{align}\label{L4.3.1}
\int\chuu{x}{k}^\ad\frac{\dif M}{\dif\tau}g\dif\tau&=0&\text{for all }x&\in\GsS\text{, }k\in\mn{1}{q},
\end{align}
and
\begin{equation}\label{L4.3.2}
\int f^\ad\frac{\dif M}{\dif\tau}g\dif\tau
\neq0.
\end{equation}
From \eqref{L4.3.1} we can derive that the \taval{\Cq} measure \((\dif M/\dif\tau)g\dif\tau\) is absolutely continuous if \(\GsS\) is an \tACs{}.
Since \(f\in\Loa{\alpha}{\si{M}}\), it follows \(\int f^\ad(\dif M/\dif\tau)g\dif\tau=0\), a contradiction to \eqref{L4.3.2}.
\end{proof}
We conclude the section with some examples of \tACs{s}.
A deeper study of this class of subsets would be of interest.
\begin{exam}\label{E4.4}
\begin{itemize}
\item[a)] If \(S\) is a compact subset of \(G\), then \(\GsS\) is an \tACs{}.
For if \(\mu\) is a regular \taval{\Cq} measure on \(\BsaG\) and \(\fsinv{\mu}(x)=0\), \(x\in\GsS\), then the continuous function \(\fsinv{\mu}\) has compact support and, hence, is integrable with respect to a Haar measure on \(G\).
Applying the inversion theorem, \tcf{}~\zitaa{MR0152834}{\cthm{1.5.1}}, and a uniqueness property of the inverse Fourier-Stieltjes transform, \tcf{}~\zitaa{MR0152834}{\cthm{1.3.6}}, we obtain that \(\mu\) is absolutely continuous.
\item[b)] Each subset of a compact abelian group is an \tACs{}.
\item[c)] If \(S\) is an \tACs{}, then \(-S\) and \(x+S\), \(x\in G\), are \tACs{s}, and if \(S\subseteq S_1\subseteq G\), then \(S_1\) is an \tACs{}.
\item[d)] If \(G=\Z\) and \(S=\N\) or \(G=\R\) and \(S=[0,\infty)\), \tresp{}, then the Theorem of F.~and M.~Riesz implies that \(S\) is an \tACs{}, \tcf{}~\zitaa{MR0152834}{\cthmss{8.2.1}{8.2.7}}.
\item[e)] A theorem of Bochner claims that the set of points of \(\Z^2\) which belong to a closed sector of the plane whose opening is larger than \(\pi\), is an \tACs{}, \tcf{}~\zitaa{MR0152834}{\cthm{8.2.5}}.
Note that for the measure \(\mu\defeq\lambda\otimes\delta_0\) on \(\Bsa{(-\pi,\pi]\times(-\pi,\pi]}\), where \(\delta_0\) denotes the Dirac measure at \(0\), one has \(\fsinv{\mu}((m,n))=0\) for all \(m\in\Z\setminus\set{0}\), \(n\in\Z\).
This shows that the lattice points of a half-plane do not form an \tACs{}.
\end{itemize}
\end{exam}
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\section{The case of an absolutely continuous measure}\label{5}
Let \(\dif M=W\dif\lambda\), where \(W\) is a \tamable{\BsaG} \taval{\Mggq} function on \(\Gamma\) integrable with respect to \(\lambda\).
For brevity, set \(\LaM\eqdef\LaW\).
It is clear that the value of \(d\) does not depend on how large the \tMec{es} are chosen.
Recall that \(P(\gamma)\) is the orthoprojection in \(\Cq\) onto \(\ran{W(\gamma)}\), \(\gamma\in\Gamma\).
If \(f\) is a \tam{\BsaG}able \taval{\Cq} function, then \(f\) and \(Pf\) are \tMe{}.
Therefore, each \tMec{} contains a function \(h\) such that
\begin{align}\label{E5.1}
h(\gamma)&\in\Ran{P(\gamma)}&\text{for all }\gamma&\in\Gamma,
\end{align}
and shrinking the \tMec{} we can and shall assume that for all functions \(h\) of the equivalence classes of \(\LaW\) relation \eqref{E5.1} is satisfied.
The goal of the present section is to derive expressions for \(d\) if \(q=1\) and \(\alpha\in\iva\) or if \(q\in\N\) and \(\alpha=2\).
Let us assume first that \(q=1\) and let us denote the scalar-valued weight function \(W\) by \(w\).
Setting \(B\defeq\setaa{\gamma\in\Gamma}{w(\gamma)\neq0}\), one has
\[
w^\mpi(\gamma)
=
\begin{cases}
1/w(\gamma),&\text{if }\gamma\in B\\
0,&\text{if }\gamma\in\Gamma\setminus B
\end{cases}
\]
and condition \eqref{E5.1} implies that if \(f\in\Law\), then \(f=0\) on \(\Gamma\setminus B\).
The distance \(d\) can be written as \[\begin{split}
d
&=\inf\setaa*{\noa{\chu{s}-t}}{t\in\T{\GsS}}\\
&=\inf\setaa*{\rk*{\int\abs{\chu{s}-t}^\alpha w\dif\lambda}^{1/\alpha}}{t\in\T{\GsS}}\\
&=\inf\setaa*{\rk*{\int\abs{\ind{B}\chu{s}-t}^\alpha w\dif\lambda}^{1/\alpha}}{t\in\ind{B}\T{\GsS}},
\end{split}\]
\(s\in S\).
For \(\alpha\in\iva\), let \(\da \defeq\alpha/(\alpha-1)\) and \(\beta\defeq1/(\alpha-1)\).
\begin{lem}\label{L5.1}
For any bounded linear functional \(\ell\) on \(\Law\), there exists \(h\in\Ldawpb \) such that
\begin{align*}
\ell(f)&=\int f h^\ad\dif\lambda,&f&\in\Law.
\end{align*}
The mapping \(\ell\mapsto h\) establishes an isometric isomorphism between the dual space of \(\Law\) and the space \(\Ldawpb \).
\end{lem}
\begin{proof}
If \(g\in\Loa{\da}{w}\), then \(\int\abs{gw}^\da \rk{w^\mpi}^\beta\dif\lambda=\int\abs{g}^\da w\dif\lambda\), which shows that the correspondence \(g\mapsto gw\), \(g\in\Loa{\da}{w}\), is an isometry from \(\Loa{\da}{w}\) into \(\Ldawpb \).
Moreover, if \(h\in\Loa{\da}{\rk{w^\mpi}^\beta}\), then \(hw^\mpi w=h\) by \eqref{E5.1} and \(\int\abs{hw^\mpi}^\da w\dif\lambda=\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda<\infty\).
Therefore, the correspondence \(g\mapsto gw\) maps \(\Loa{\da}{w}\) onto \(\Loa{\da}{\rk{w^\mpi}^\beta}\), and the lemma follows from the well known description of the dual space of \(\Law\), \tcf{}~\rlem{L2.1}.
\end{proof}
The preceding lemma shows that if \(f\in\Law\) and \(h\in\Ldawpb \) satisfies \eqref{E5.1}, then
\[
\int\abs*{fh^\ad}\dif\lambda
\leq\noa{f}\ek*{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{1/\da}.
\]
It follows that under condition \eqref{E5.1} the set
\[
\D
\defeq\setaa*{h\in\LoA{\da}{\rk{w^\mpi}^\beta}}{\fsinv{\rk{h^\ad}}(x)=0\text{ for all }x\in\GsS}
\]
is defined correctly and that \(\int fh^\ad\dif\lambda=0\) if \(f\in\cl{\T{\GsS}}\) and \(h\in\D\).
Taking into account a general approximation result in Banach spaces, \tcf{}~\zitaa{MR0268655}{\cthm{7.2}}, we can conclude that the following assertion is true.
\begin{lem}\label{L5.2}
For any \(s\in S\), the distance \(d\) is equal to \(\sup\setaa{\abs{\fsinv{\rk{h^\ad}}(s)}}{h\in\D\text{ and } \int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda\leq1}\).
\end{lem}
Let \(\Ds\defeq\setaa{h\in\D}{\fsinv{\rk{h^\ad}}(s)=1}\), \(s\in S\).
\begin{thm}\label{T5.3}
Let \(q=1\), \(\alpha\in\iva\), and \(s\in S\).
Then
\begin{equation}\label{E5.2}
d
=\inf\setaa*{\noa{\chu{s}-t}}{t\in\T{\GsS}}
=\sup\setaa*{\ek*{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{-1/\da}}{h\in\Ds},
\end{equation}
with the convention, that the right-hand side of \eqref{E5.2} is assumed to be \(0\) if \(\Ds\) is empty.
\end{thm}
\begin{proof}
Define on the linear space \(\D\) two positive homogeneous and non-negative functionals \(\F\) and \(\G\) by \(\F(h)\defeq\ek{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{1/\da}\) and \(\G(h)\defeq\abs{\fsinv{\rk{h^\ad}}(s)}\), \(h\in\D\).
If \(\G(h)=0\) for all \(h\in\D\), then the left-hand side of \eqref{E5.2} is \(0\) by \rlem{L5.2} and the right-hand side equals \(0\) by the convention made.
If \(\G(h)\neq0\) for some \(h\in\D\), then \rlem{L5.2} and a well known duality relation, \tcf{}~\zitaa{MR0224158}{\clem{7.1}}, imply that the distance $d$ is equal to \(d=\sup\setaa{\G(h)}{h\in\D\text{ and }\F(h)\leq1}=\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)\geq1}}^\inv\).
Since \(\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)\geq1}}^\inv={\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)=1}}^\inv}=\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\fsinv{\rk{h^\ad}}(s)=1}}^\inv=\sup\setaa{\ek{\F(h)}^\inv}{h\in\Ds}\), the theorem is proved.
\end{proof}
\begin{cor}\label{C5.4}
The set \(\T{\GsS}\) is dense in \(\Law\) if and only if there does not exist a function \(h\in\D\setminus\set{0}\) such that \(\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda<\infty\).
\end{cor}
\begin{proof}
Note that \(h\in\D\setminus\set{0}\) if and only if \(h\in\D\) and there exists \(s\in S\) such that \(\fsinv{\rk{h^\ad}}(s)\neq0\), which gives \(\D\setminus\set{0}=\bigcup\setaa{a\Ds}{a\in\C\setminus\set{0}\text{, }s\in S}\).
If \(\delta\) denotes the right-hand side of \eqref{E5.2}, we obtain the following chain of equivalences: \(\T{\GsS}\) is dense in \(\Law\)\tarlr{}\(d=0\) for all \(s\in S\)\tarlr{}\(\delta=0\) for all \(s\in S\)\tarlr{}\(\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda=\infty\) for all \(h\in\D\setminus\set{0}\).
\end{proof}
If \(q\) is an arbitrary positive integer, an analogous result to that of \rlem{L5.1} is not true in general.
However, for \(\alpha=2\), the above method can be extended.
We briefly sketch the main steps.
\begin{description}
\item[Step 1:] The correspondence \(\ell\mapsto h\), defined by \(\ell(f)=\int h^\ad f\dif\lambda\), \(f\in\Loa{2}{W}\), establishes an isometric isomorphism between the dual space of \(\Loa{2}{W}\) and the space \(\Loa{2}{W^\mpi}\).
\item[Step 2:] \(\int\abs{h^\ad f}\dif\lambda\leq\noq{f}\ek{\int h^\ad W^\mpi h\dif\lambda}^{1/2}\) for all \(f\in\Loa{2}{W}\) and \(h\in\Loa{2}{W^\mpi}\) satisfying \eqref{E5.1}.
\item[Step 3:] The set \(\Dt\defeq\setaa{h\in\Loa{2}{W^\mpi}}{\fsinv{\rk{h^\ad}}(x)=0\text{ for }x\in\GsS}\) is defined correctly and \(\int h^\ad f\dif\lambda=0\) for all \(f\in\Loa{2}{W}\) and \(h\in\Loa{2}{W^\mpi}\) satisfying \eqref{E5.1}.
\item[Step 4:] For all \(s\in S\) and \(k\in\mn{1}{q}\), the distance \(d\) is equal to \(\sup\setaa{\abs{\fsinv{\rk{h^\ad}}(s)\eu{k}}}{h\in\Dt\text{ and } \int h^\ad W^\mpi h\dif\lambda\leq1}\).
\item[Step 5:] For \(s\in S\) and \(k\in\mn{1}{q}\), set \(\Dtsk\defeq\setaa{h\in\Dt}{\fsinv{\rk{h^\ad}}(s)\eu{k}=1}\).
Introducing two positive homogeneous and non-negative functionals \(\Ft\) and \(\Gt\) on \(\Dt\) by \(\Ft(h)\defeq\ek{\int h^\ad W^\mpi h\dif\lambda}^{1/2}\) and \(\Gt(h)\defeq\abs{\fsinv{\rk{h^\ad}}(s)\eu{k}}\), \(h\in\Dt\), similarly to the proof of \rthm{T5.3} one can derive the following assertion.
\end{description}
\begin{thm}\label{T5.5}
Let \(q\in\N\) and \(s\in S\).
Then
\begin{equation}\label{E5.3}
d
=\inf\setaa*{\noq{\chuu{s}{k}-t}}{t\in\T{\GsS}}
=\sup\setaa*{\ek*{\int h^\ad W^\mpi h\dif\lambda}^{-1/2}}{h\in\Dtsk},
\end{equation}
where the right-hand side of \eqref{E5.3} is to be interpreted as \(0\) if \(\Dtsk\) is empty.
\end{thm}
\begin{cor}[\tcf{}~\zitaa{MR0426126}{\ccor{3.16}}]\label{C5.6}
Let \(q\in\N\).
The set \(\T{\GsS}\) is dense in \(\Loa{2}{W}\) if and only if there does not exist a function \(h\in\Dt\setminus\set{0}\) such that \(\int h^\ad W^\mpi h\dif\lambda<\infty\).
\end{cor}
\noindent
\begin{minipage}{0.5\textwidth}
Universit\"at Leipzig\\
Fakult\"at f\"ur Mathematik und Informatik\\
PF~10~09~20\\
D-04009~Leipzig
\end{minipage}
\begin{minipage}{0.49\textwidth}
\begin{flushright}
\texttt{
[email protected]\\
[email protected]
}
\end{flushright}
\end{minipage}
\end{document}
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\begin{document}
\title{Condition Number of\ Full Rank Linear least-squares Solutions}
\begin{abstract}
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.
\end{abstract}
\begin{keywords}
linear least-squares, condition number, applications of functional analysis, nuclear norm, trace norm
\end{keywords}
\begin{AMS}
65F35, 62J05, 15A60
\end{AMS}
\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{JOSEPH F. GRCAR}{LINEAR LEAST SQUARES SOLUTION}
\section {Introduction}
\subsection {Purpose}
Linear least-squares problems, in the form of statistical regression analyses, are a basic tool of investigation in both the physical and the social sciences, and consequently they are an important computation.
This paper develops a single methodology that determines tight lower and upper estimates of condition numbers for several problems involving linear least-squares. The condition numbers are with respect to the matrices in the problems and scaled $2$-norms. The problems are: orthogonal projections and least-squares residuals \citep {Grcar2010f}, minimum $2$-norm solutions of underdetermined equations \citep {Grcar2010b}, and in the present case, the solution of overdetermined equations
\begin {equation}
\label {eqn:LLS}
x = \arg \min_u \| b - A u \|_2 \quad \Rightarrow \quad A^t A x = A^t b \, ,
\end {equation}
where $A$ is an $m \times n$ matrix of full column rank (hence, $m \ge n$). Some presentations of error bounds contain formulas that can severely overestimate the condition number, including the SIAM documentation for the LAPACK software.
This introduction provides some background material. Section \ref {sec:definition} discusses definitions of condition numbers. Section \ref {sec:brief} describes the estimate and provides an example; this material is appropriate for presentation in class. Section \ref {sec:derivation} proves that the condition number varies from the estimate within a factor of $\sqrt 2$. The derivation relies on a formula for the nuclear norm of a matrix. (This norm is the sum of the singular values including multiplicities, and is also known as the trace norm.) Section \ref {sec:comparison} examines overestimates in the literature. Section \ref {sec:advanced} evaluates the nuclear norm of rank $2$ matrices (lemma \ref {lem:rank2}).
\subsection {Prior Work}
Ever since \citet {Legendre1805-HM} and \citet {Gauss1809-HM} invented the method of least-squares, the problems had been solved by applying various forms of elimination to the normal equations, $A^t A x = A^t b$ in equation (\ref {eqn:LLS}). Instead, \citet {Golub1965} suggested applying Householder transformations directly to $A$, which removed the need to calculate $A^t A$. However, \citet [p.\ 144] {GolubWilkinson1966} reported that $A^tA$ was still ``relevant to some extent'' to the accuracy of the calculation because they found that $A^tA$ appears in a bound on perturbations to $x$ that are caused by perturbations to $A$. Their discovery was ``something of a shock'' \citep [p.\ 241] {vanderSluis1975}.
The original error bound of \citet [p.\ 144, eqn.\ 43] {GolubWilkinson1966} was difficult to interpret because of an assumed scaling for the problem. \citet [pp.\ 15, 17, top] {Bjorck1967a} derived a bound by the augmented matrix approach that was suggested to him by Golub. \citet [pp.\ 224--226] {Wedin1973} re-derived the bound from his study of the matrix pseudoinverse and exhibited a perturbation to the matrix that attains the leading term. Van der Sluis (1975, p.\ 251, eqn.\ 5.8) also derived Bj\"orck's bound and introduced a simplification of the formula and a geometric interpretation of the leading term. \citet [p.\ 31, eqn.\ 1.4.28] {Bjorck1996} later followed Wedin in basing the derivation of his bound on the pseudoinverse. \citet [p.\ 1189, eqn.\ 2.4 and line --6] {Malyshev2003} derived a lower bound for the condition number thereby proving that his formula and the coefficient in Bj\"orck's bound are quantifiably tight estimates of the spectral condition number. In contrast, condition numbers with respect to Frobenius norms have exact formulas that have been given in various forms by \citet {Geurts1982}, \citet {Gratton1996}, and \citet {Malyshev2003}.
\section {Condition numbers}
\label {sec:definition}
\subsection {Error bounds and definitions of condition numbers}
The oldest way to derive perturbation bounds is by differential calculus. If $y = f(x)$ is a vector valued function of the vector $x$ whose partial derivatives are continuous, then the partial derivatives give the best estimate of the change to $y$ for a given change to $x$
\begin {equation}
\label {eqn:approximation-1}
\Dy = f(x + \Dx) - f(x) \approx J_f (x) \, \Dx
\end {equation}
where $J_f (x)$ is the Jacobian matrix of the partial derivatives of $y$ with respect to $x$. The magnitude of the error in the first order approximation (\ref {eqn:approximation-1}) is bounded by Landau's little $o ( \| \Dx \| )$ for all sufficiently small $\| \Dx \|$.\footnote {The continuity of the partial derivatives establishes the existence of the Fr\'echet derivative and its representation by the Jacobian matrix. The definition of the Fr\'echet derivative is responsible for the error in equation (\ref {eqn:approximation-1}) being $o ( \| \Dx \| )$. The order of the error terms is independent of the norm because all norms for finite dimensional spaces are equivalent \citep [p.\ 54, thm.\ 1.7] {Stewart1990}.} Thus $J_f (x) \, \Dx$ is the unique linear approximation to $\Dy$ in the vicinity of $x$.\footnote {Any other linear function added to $J_f (x) \, \Dx$ differs from $\Dy$ by ${\mathcal O} (\| \Dx \|)$ and therefore does not provide a $o ( \| \Dx \| )$ approximation.} Taking norms produces a perturbation bound,
\begin {equation}
\label {eqn:calculus-1}
\| \Dy \| \le \| J_f (x) \| \, \| \Dx \| + o (\| \Dx \|) \, .
\end {equation}
Equation (\ref {eqn:calculus-1}) is the smallest possible bound on $\| \Dy \|$ in terms of $\| \Dx \|$ provided the norm for the Jacobian matrix is induced from the norms for $\Dy$ and $\Dx$. In this case for each $x$ there is some $\Dx$, which is nonzero but may be chosen arbitrarily small, so the bound (\ref {eqn:calculus-1}) is attained to within the higher order term, $o (\| \Dx \|)$. There may be many ways to define condition numbers, but because equation (\ref {eqn:calculus-1}) is the smallest possible bound, any definition of a condition number for use in bounds equivalent to (\ref {eqn:calculus-1}) must arrive at the same value, $\chi_y (x) = \| J_f (x) \|$.\footnote {A theory of condition numbers in terms of Jacobian matrices was developed by \citet [p.\ 292, thm.\ 4] {Rice1966}. Recent presentations of the formula $\chi_y (x) = \| J_f (x) \|$ are given by \citet [p.\ 44] {Chatelin1996}, \citet [p.\ 27] {Deuflhard2003}, \citet [p.\ 39] {Quarteroni2000}, and \citet [p.\ 90] {Trefethen1997}.} The matrix norm may be too complicated to have an explicit formula, but tight estimates can be derived as in this paper.
\subsection {One or separate condition numbers}
\label {sec:separate}
Many problems depend on two parameters $u$, $v$ which may consist of the entries of a matrix and a vector (for example). In principle it is possible to treat the parameters altogether.\footnote {As will be seen in table \ref {tab:LLS}, \citet {Gratton1996} derived a joint condition number of the least-squares solution with respect to a Frobenius norm of the matrix and vector that define the problem.} A condition number for $y$ with respect to joint changes in $u$ and $v$ requires a common norm for perturbations to both. Such a norm is
\begin {equation}
\label {eqn:joint-norm}
\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \, .
\end {equation}
A single condition number then follows that appears in an optimal error bound,
\begin {equation}
\label {eqn:single}
\| \Dy \| \le \| J_f (u, v) \| \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) .
\end {equation}
The value of the condition number is again $\chi_y(u, v) = \| J_f (u, v) \|$ where the matrix norm is induced from the norm for $\Dy$ and the norm in equation (\ref {eqn:joint-norm}).
Because $u$ and $v$ may enter into the problem in much different ways, it is customary to treat each separately. This approach recognizes that the Jacobian matrix is a block matrix
\begin {displaymath}
J_f (u, v) = \onetwo {J_{f_1} (u)} {J_{f_2} (v)}
\end {displaymath}
where the functions $f_1 (u) = f(u, v)$ and $f_2(v) = f(u, v)$ have $v$ and $u$ fixed, respectively.
The first order differential approximation (\ref {eqn:approximation-1}) is unchanged but is rewritten with separate terms for $u$ and $v$,
\begin {equation}
\label {eqn:approximation-2}
\Dy \approx J_{f_1} (u) \, \Du + J_{f_2} (v) \, \Dv \, .
\end {equation}
Bound (\ref {eqn:single}) then can be weakened by applying norm inequalities,
\begin {eqnarray}
\nonumber
\| \Dy \|& \le& \| J_{f_1} (u) \Du + J_{f_2} (v) \Dv \| + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right)\\
\noalign {
}
\label {eqn:double}
& \le& \left( \strut \| J_{f_1} (u) \| + \| J_{f_2} (v) \| \right) \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\}\\
\nonumber
&& \hspace* {12em} {} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) \, .
\end {eqnarray}
The coefficients $\chi_y(u) = \| J_{f_1} (u) \|$ and $\chi_y(v) = \| J_{f_2} (v) \|$ are the separate condition numbers of $y$ with respect to $u$ and $v$, respectively.
These two different approaches lead to error bounds (\ref {eqn:single}, \ref {eqn:double}) that differ by at most a factor of $2$ because it can be shown \citep {Grcar2010f}
\begin {equation}
\label {eqn:sum-6}
{\chi_y (u) + \chi_y (v) \over 2} \le \chi_y (u, v) \le \chi_y (u) + \chi_y (v) \, .
\end {equation}
Thus, for the purpose of deriving tight estimates of joint condition numbers, it suffices to consider $\chi_y (u)$ and $\chi_y (v)$ separately.
\section {Conditioning of the least-squares solution}
\label {sec:brief}
\subsection {Reason for considering matrices of full column rank}
\label {sec:reason}
The linear least-squares problem (\ref {eqn:LLS}) does not have an unique solution when $A$ does not have full column rank. A specific $x$ can be chosen such as the one of minimum norm. However, small changes to $A$ can still produce large changes to $x$.\footnote {If $A$ does not have full column rank, then for every nonzero vector $z$ in the right null space of the matrix, $(A + b z^t) (z / z^t z) = b$. Thus, a change to the matrix of norm $\| b \|_2 \, \|z \|_2$ changes the solution from $A^\dag b$ to $z /\| z \|_2^2$.} In other words, a condition number of $x$ with respect to rank deficient $A$ does not exist or is ``infinite.'' Perturbation bounds in the rank deficient case can be found by restricting changes to the matrix, for which see \citet [p.\ 30, eqn.\ 1.4.26] {Bjorck1996} and \citet [p.\ 157, eqn.\ 5.3] {Stewart1990}. That theory is beyond the scope of the present discussion.
\subsection {The condition numbers}
\label {sec:brief-1}
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\subsection {One or separate condition numbers}
\label {sec:separate}
Many problems depend on two parameters $u$, $v$ which may consist of the entries of a matrix and a vector (for example). In principle it is possible to treat the parameters altogether.\footnote {As will be seen in table \ref {tab:LLS}, \citet {Gratton1996} derived a joint condition number of the least-squares solution with respect to a Frobenius norm of the matrix and vector that define the problem.} A condition number for $y$ with respect to joint changes in $u$ and $v$ requires a common norm for perturbations to both. Such a norm is
\begin {equation}
\label {eqn:joint-norm}
\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \, .
\end {equation}
A single condition number then follows that appears in an optimal error bound,
\begin {equation}
\label {eqn:single}
\| \Dy \| \le \| J_f (u, v) \| \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) .
\end {equation}
The value of the condition number is again $\chi_y(u, v) = \| J_f (u, v) \|$ where the matrix norm is induced from the norm for $\Dy$ and the norm in equation (\ref {eqn:joint-norm}).
Because $u$ and $v$ may enter into the problem in much different ways, it is customary to treat each separately. This approach recognizes that the Jacobian matrix is a block matrix
\begin {displaymath}
J_f (u, v) = \onetwo {J_{f_1} (u)} {J_{f_2} (v)}
\end {displaymath}
where the functions $f_1 (u) = f(u, v)$ and $f_2(v) = f(u, v)$ have $v$ and $u$ fixed, respectively.
The first order differential approximation (\ref {eqn:approximation-1}) is unchanged but is rewritten with separate terms for $u$ and $v$,
\begin {equation}
\label {eqn:approximation-2}
\Dy \approx J_{f_1} (u) \, \Du + J_{f_2} (v) \, \Dv \, .
\end {equation}
Bound (\ref {eqn:single}) then can be weakened by applying norm inequalities,
\begin {eqnarray}
\nonumber
\| \Dy \|& \le& \| J_{f_1} (u) \Du + J_{f_2} (v) \Dv \| + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right)\\
\noalign {
}
\label {eqn:double}
& \le& \left( \strut \| J_{f_1} (u) \| + \| J_{f_2} (v) \| \right) \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\}\\
\nonumber
&& \hspace* {12em} {} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) \, .
\end {eqnarray}
The coefficients $\chi_y(u) = \| J_{f_1} (u) \|$ and $\chi_y(v) = \| J_{f_2} (v) \|$ are the separate condition numbers of $y$ with respect to $u$ and $v$, respectively.
These two different approaches lead to error bounds (\ref {eqn:single}, \ref {eqn:double}) that differ by at most a factor of $2$ because it can be shown \citep {Grcar2010f}
\begin {equation}
\label {eqn:sum-6}
{\chi_y (u) + \chi_y (v) \over 2} \le \chi_y (u, v) \le \chi_y (u) + \chi_y (v) \, .
\end {equation}
Thus, for the purpose of deriving tight estimates of joint condition numbers, it suffices to consider $\chi_y (u)$ and $\chi_y (v)$ separately.
\section {Conditioning of the least-squares solution}
\label {sec:brief}
\subsection {Reason for considering matrices of full column rank}
\label {sec:reason}
The linear least-squares problem (\ref {eqn:LLS}) does not have an unique solution when $A$ does not have full column rank. A specific $x$ can be chosen such as the one of minimum norm. However, small changes to $A$ can still produce large changes to $x$.\footnote {If $A$ does not have full column rank, then for every nonzero vector $z$ in the right null space of the matrix, $(A + b z^t) (z / z^t z) = b$. Thus, a change to the matrix of norm $\| b \|_2 \, \|z \|_2$ changes the solution from $A^\dag b$ to $z /\| z \|_2^2$.} In other words, a condition number of $x$ with respect to rank deficient $A$ does not exist or is ``infinite.'' Perturbation bounds in the rank deficient case can be found by restricting changes to the matrix, for which see \citet [p.\ 30, eqn.\ 1.4.26] {Bjorck1996} and \citet [p.\ 157, eqn.\ 5.3] {Stewart1990}. That theory is beyond the scope of the present discussion.
\subsection {The condition numbers}
\label {sec:brief-1}
This section summarizes the results and presents an example. Proofs are in section \ref {sec:derivation}. It is assumed that $A$ has full column rank and the solution $x$ of the least-squares problem (\ref {eqn:LLS}) is not zero. The solution is proved to have a condition number $\chi_x(A)$ with respect to $A$ within the limits,
\begin {equation}
\label {eqn:chixA}
\fbox {$\boldkappa \, \sqrt { \strut [\vds \tan (\boldtheta)]^2 + 1} $} \; \le \; \chi_x (A) \; \le \; \fbox {$\boldkappa [\vds \tan (\boldtheta) + 1]$} \, ,
\end {equation}
where $\boldkappa$, $\vds$, and $\boldtheta$ are defined below; they are bold to emphasize they are the values in the tight estimates of the condition number. There is also condition number with respect to $b$,
\begin {equation}
\label {eqn:chixb}
\chi_x (b) = \fbox {$\vds \sec (\boldtheta)$} \, .
\end {equation}
These condition numbers $\chi_x(A)$ and $\chi_x (b)$ are for measuring the perturbations to $A$, $b$, and $x$ by the following scaled $2$-norms,
\begin {equation}
\label {eqn:specific-scaled-norms}
{\| \DA \|_2 \over \| A \|_2} \, ,
\qquad
{\| \Db \|_2 \over \| b \|_2} \, ,
\qquad
{\| \Dx \|_2 \over \| x \|_2} \, .
\end {equation}
Like equation (\ref {eqn:double}), the two condition numbers appear in error bounds of the form,\footnote {The constant denominators $\| A \|_2$ and $\| b \|_2$ could be discarded from the $o$ terms because only the order of magnitude of the terms is pertinent.}
\begin {equation}
\label {eqn:error-bound}
{\| \Dx \|_2 \over \| x \|_2} \le \chi_x (A) {\| \DA \|_2 \over \| A \|_2} + \chi_x (b) {\| \Db \|_2 \over \| b \|_2} + o \left( \max \left\{ {\| \DA \|_2 \over \| A \|_2}, \, {\| \Db \|_2 \over \| b \|_2} \right\} \right) ,
\end {equation}
where $x + \Dx$ is the solution of the perturbed problem,
\begin {equation}
\label {eqn:perturbed-problem}
x + \Dx = \arg \min_u \| (b + \Db) - (A + \DA) u \|_2 \, .
\end {equation}
The quantities $\boldkappa$, $\vds$, and $\boldtheta$ in the formulas (\ref {eqn:chixA}, \ref {eqn:chixb}) are
\begin {equation}
\label {eqn:three}
\boldkappa = {\| A \|_2 \over \sigmamina} \qquad
\vds = {\| A x \|_2 \over \| x \|_2 \, \sigmamina} \qquad
\tan (\boldtheta) = {\| r \|_2 \over \| A x \|_2}
\end {equation}
where $\boldkappa$ is the spectral matrix condition number of $A$ ($\sigmamin$ is the smallest singular value of $A$), $\vds$ is van der Sluis's ratio between $1$ and $\boldkappa$,\footnote {The formulas of \citet [p.\ 251] {vanderSluis1975} contain in his notation $R(x_0) / \sigma_n$, which is the present $\vds$.} $\boldtheta$ is the angle between $b$ and $\col (A)$,\footnote {The notation $\col (A)$ is the column space of $A$.} and $r = b - Ax$ is the least-squares residual.
\begin {enumerate}
\item $\boldkappa$ depends only on the extreme singular values of $A$.
\item $\boldtheta$ depends only on the ``angle of attack'' of $b$ with respect to $\col (A)$.
\item If $A$ is fixed, then $\vds$ depends on the orientation of $b$ to $\col (A)$ but not on $\boldtheta$.\footnote {Because $A$ has full column rank, $Ax$ and $x$ can only vary proportionally when their directions are fixed.}
\end {enumerate}
Please refer to Figure \ref {fig:schematic} as needed.
If $\col (A)$ is fixed, then $\boldkappa$ and $\vds$ depend only on the singular values of $A$, and $\boldtheta$ depends only on the orientation of $b$. Thus, $\boldkappa$ and $\boldtheta$ are separate sources of ill-conditioning for the solution. If $Ax$ has comparatively large components in singular vectors corresponding to the largest singular values of $A$, then $\vds \approx \boldkappa$ and the condition number $\chi_x (A)$ depends on $\boldkappasquared$ which was the discovery of \citet {GolubWilkinson1966}. Otherwise, $\boldkappasquared$ ``plays no role'' \citep [p.\ 251] {vanderSluis1975}.
\begin {figure}
\centering
\includegraphics [scale=1] {schematic}
\caption {Schematic of the least-squares problem, the projection $Ax$, and the angle $\boldtheta$ between $Ax$ and $b$.}
\label {fig:schematic}
\end {figure}
\subsection {Conditioning example}
\label {sec:example}
This example illustrates the independent effects of $\boldkappa$, $\vds$, and $\boldtheta$ on $\chi_x (A)$. It is based on the example of \citet [p.\ 238] {Golub1996}. Let
\begin {displaymath}
A = \left[ \begin {array} {c c} 1& 0\\ 0& \alpha\\ 0& 0\\ \end {array}
\right] ,
\quad
b = \left[ \begin {array} {c} \beta \cos (\phi)\\ \beta \sin (\phi) \\ 1 \end {array} \right] \, ,
\quad
\DA = \left[ \begin {array} {c c} 0& 0\\ 0& 0\\ 0& \epsilon \end {array}
\right] .
\end {displaymath}
where $0 < \epsilon \ll \alpha, \beta < 1$.
In this example,
\begin {displaymath}
x = \left[ \begin {array} {c} \beta \cos (\phi) \\ {\beta \vphantom {(} \over \vphantom {(} \alpha} \sin (\phi) \\ \end {array} \right] ,
\quad
r = \left[ \begin {array} {c} 0\\ 0\\ 1 \end {array} \right] ,
\quad
x + \Dx = \left[ \begin {array} {c} \beta \cos (\phi) \\[0.1ex] {\epsilon + \alpha \beta \sin (\phi) \vphantom {(} \over \vphantom {(}\alpha^2 + \epsilon^2} \\ \end {array} \right] \, .
\end {displaymath}
The three terms in the condition number are
\begin {displaymath}
\boldkappa = {1 \over \alpha} \, , \qquad
\vds = {1 \over \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} \, , \qquad
\tan (\boldtheta) = {1 \over \beta} \, .
\end {displaymath}
These values can be independently manipulated by choosing $\alpha$, $\beta$, and $\phi$. The tight upper bound for the condition number is
\begin {displaymath}
\chi_x (A) \le {1 \over \alpha} \left ( {1 \over \beta \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + 1 \right) \, .
\end {displaymath}
The relative change to the solution of the example
\begin {displaymath}
{\| \Dx \|_2 \over \| x \|_2}
= {\epsilon \over \alpha \, \beta \, \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + {\mathcal O} (\epsilon^2) \, .
\end {displaymath}
is close to the bound given by the condition number estimate and the relative change to $A$.
\section {Derivation of the condition numbers}
\label {sec:derivation}
\subsection {Notation}
The formula for the Jacobian matrix $J_x (b)$ of the solution $x = (A^t A)^{-1} b$ with respect to $b$ is clear.\footnote {The notation of section \ref {sec:separate} would introducing a name, $f_2$, for the function by which $x$ varies with $b$ when $A$ is held fixed, $x = f_2 (b)$, so that the notation for the Jacobian matrix is then $J_{f_2} (b)$. This pedantry will be discarded here to write $J_x(b)$ with $A$ held fixed; and similarly for $J_x(A)$ with $b$ held fixed.} For derivatives with respect to the entries of $A$, it is necessary to use the ``$\Vector$'' construction to order the matrix entries into a column vector; $\Vector (B)$ is the column of entries $B_{i,j}$ with $i,j$ in co-lexicographic order.\footnote {The alternative to placing the entries of matrices into column vectors is to use more general linear spaces and the Fr\'echet derivative. That approach seems unnecessarily abstract because the spaces have finite dimension.} The approximation is then
\begin {equation}
\label {eqn:total-differential}
\Dx = \JxA \, \Vector (\DA) + \Jxb \, \Db + \mbox {higher order terms}
\end {equation}
and upon taking norms
\begin {equation}
\label {eqn:differential-bound}
\| \Dx \| \le \underbrace {\| \JxA \|}_{\displaystyle \chi_x (A)} \, \| \DA \| + \underbrace {\displaystyle \| \Jxb \|}_{\displaystyle \chi_x (b)} \, \| \Db \| + o( \max \{ \| \DA \|, \, \| \Db \| \} ) \, ,
\end {equation}
where it is understood the norms on the two Jacobian matrices are induced from the following norms for $\DA$, $\Db$, and $\Dx$.
\subsection {Choice of Norms}
\label {sec:norms}
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Please refer to Figure \ref {fig:schematic} as needed.
If $\col (A)$ is fixed, then $\boldkappa$ and $\vds$ depend only on the singular values of $A$, and $\boldtheta$ depends only on the orientation of $b$. Thus, $\boldkappa$ and $\boldtheta$ are separate sources of ill-conditioning for the solution. If $Ax$ has comparatively large components in singular vectors corresponding to the largest singular values of $A$, then $\vds \approx \boldkappa$ and the condition number $\chi_x (A)$ depends on $\boldkappasquared$ which was the discovery of \citet {GolubWilkinson1966}. Otherwise, $\boldkappasquared$ ``plays no role'' \citep [p.\ 251] {vanderSluis1975}.
\begin {figure}
\centering
\includegraphics [scale=1] {schematic}
\caption {Schematic of the least-squares problem, the projection $Ax$, and the angle $\boldtheta$ between $Ax$ and $b$.}
\label {fig:schematic}
\end {figure}
\subsection {Conditioning example}
\label {sec:example}
This example illustrates the independent effects of $\boldkappa$, $\vds$, and $\boldtheta$ on $\chi_x (A)$. It is based on the example of \citet [p.\ 238] {Golub1996}. Let
\begin {displaymath}
A = \left[ \begin {array} {c c} 1& 0\\ 0& \alpha\\ 0& 0\\ \end {array}
\right] ,
\quad
b = \left[ \begin {array} {c} \beta \cos (\phi)\\ \beta \sin (\phi) \\ 1 \end {array} \right] \, ,
\quad
\DA = \left[ \begin {array} {c c} 0& 0\\ 0& 0\\ 0& \epsilon \end {array}
\right] .
\end {displaymath}
where $0 < \epsilon \ll \alpha, \beta < 1$.
In this example,
\begin {displaymath}
x = \left[ \begin {array} {c} \beta \cos (\phi) \\ {\beta \vphantom {(} \over \vphantom {(} \alpha} \sin (\phi) \\ \end {array} \right] ,
\quad
r = \left[ \begin {array} {c} 0\\ 0\\ 1 \end {array} \right] ,
\quad
x + \Dx = \left[ \begin {array} {c} \beta \cos (\phi) \\[0.1ex] {\epsilon + \alpha \beta \sin (\phi) \vphantom {(} \over \vphantom {(}\alpha^2 + \epsilon^2} \\ \end {array} \right] \, .
\end {displaymath}
The three terms in the condition number are
\begin {displaymath}
\boldkappa = {1 \over \alpha} \, , \qquad
\vds = {1 \over \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} \, , \qquad
\tan (\boldtheta) = {1 \over \beta} \, .
\end {displaymath}
These values can be independently manipulated by choosing $\alpha$, $\beta$, and $\phi$. The tight upper bound for the condition number is
\begin {displaymath}
\chi_x (A) \le {1 \over \alpha} \left ( {1 \over \beta \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + 1 \right) \, .
\end {displaymath}
The relative change to the solution of the example
\begin {displaymath}
{\| \Dx \|_2 \over \| x \|_2}
= {\epsilon \over \alpha \, \beta \, \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + {\mathcal O} (\epsilon^2) \, .
\end {displaymath}
is close to the bound given by the condition number estimate and the relative change to $A$.
\section {Derivation of the condition numbers}
\label {sec:derivation}
\subsection {Notation}
The formula for the Jacobian matrix $J_x (b)$ of the solution $x = (A^t A)^{-1} b$ with respect to $b$ is clear.\footnote {The notation of section \ref {sec:separate} would introducing a name, $f_2$, for the function by which $x$ varies with $b$ when $A$ is held fixed, $x = f_2 (b)$, so that the notation for the Jacobian matrix is then $J_{f_2} (b)$. This pedantry will be discarded here to write $J_x(b)$ with $A$ held fixed; and similarly for $J_x(A)$ with $b$ held fixed.} For derivatives with respect to the entries of $A$, it is necessary to use the ``$\Vector$'' construction to order the matrix entries into a column vector; $\Vector (B)$ is the column of entries $B_{i,j}$ with $i,j$ in co-lexicographic order.\footnote {The alternative to placing the entries of matrices into column vectors is to use more general linear spaces and the Fr\'echet derivative. That approach seems unnecessarily abstract because the spaces have finite dimension.} The approximation is then
\begin {equation}
\label {eqn:total-differential}
\Dx = \JxA \, \Vector (\DA) + \Jxb \, \Db + \mbox {higher order terms}
\end {equation}
and upon taking norms
\begin {equation}
\label {eqn:differential-bound}
\| \Dx \| \le \underbrace {\| \JxA \|}_{\displaystyle \chi_x (A)} \, \| \DA \| + \underbrace {\displaystyle \| \Jxb \|}_{\displaystyle \chi_x (b)} \, \| \Db \| + o( \max \{ \| \DA \|, \, \| \Db \| \} ) \, ,
\end {equation}
where it is understood the norms on the two Jacobian matrices are induced from the following norms for $\DA$, $\Db$, and $\Dx$.
\subsection {Choice of Norms}
\label {sec:norms}
Equation (\ref {eqn:differential-bound}) applies for any choice of norms. In theoretical numerical analysis especially for least-squares problems the spectral norm is preferred. For $2$-norms the matrix condition number of $A^t A$ is the square of the matrix condition number of $A$. The norms used in this paper are thus,
\begin {equation}
\label {eqn:norms}
\| \Vector (\DA) \|_\scaleA = {\| \DA \|_2 \over \scaleA} \, ,
\qquad
\| \Db \|_\scaleB = {\| \Db \|_2 \over \scaleB} \, ,
\qquad
\| \Dx \|_\scaleX = {\| \Dx \|_2 \over \scaleX} \, ,
\end {equation}
where $\scaleA$, $\scaleB$, $\scaleX$ are constant scale factors. These formulas define norms for $m \times n$ matrices, for $m$ vectors, and for $n$ vectors. The scaling makes the size of the changes relative to the problem of interest. The scaling used in equations (\ref {eqn:chixA}--\ref {eqn:specific-scaled-norms}) is
\begin {equation}
\label {eqn:scale-factors}
\scaleA = \| A \|_2 \, ,
\qquad
\scaleB = \| b \|_2 \, ,
\qquad
\scaleX = \| x \|_2 \, .
\end {equation}
\subsection {Condition number of {\itshape x\/} with respect to {\itshape b\/}}
\label {sec:conditionwrtb}
From $x = (A^t A)^{-1} A^t b$ follows $\Jxb = (A^t A)^{-1} A^t$ and then for the scaling of equation (\ref {eqn:scale-factors})
\begin {equation}
\label {eqn:conditionwrtb}
\renewcommand {\arraycolsep} {0.125em}
\begin {array} {r c l}
\| \Jxb \|&
=&
\displaystyle \max_{\Db} {\| \Jxb \, \Db \|_\scaleA \over \| \Db \|_\scaleB}
= \displaystyle \max_{\Db} {\displaystyle \left( {\| (A^t A)^{-1} A^t \Db \|_2 \over \scaleX} \right) \over \displaystyle \left( {\| \Db \|_2 \over \scaleB} \right)}
= {\scaleB \over \scaleX \, \sigmamina}\\
&
=&
\displaystyle {\| b \|_2 \over \| A x \|_2} {\| A x \|_2 \over \| x \|_2 \, \sigmamina} = \sec (\boldtheta) \, \vds \, .
\end {array}
\end {equation}
\subsection {Condition number of {\itshape x\/} with respect to {\itshape A\/}}
\label {sec:begin}
The Jacobian matrix $\JxA$ is most easily calculated from the total differential of the identity $F = A^t (b - A x) = 0$ with respect to $A$ and $x$, which is $J_F [\Vector (A)] \, \Vector (dA) + J_F (x) \, dx = 0$. Hence
\begin {equation}
\label {eqn:hence}
dx = \underbrace {- [J_F (x)]^{-1} J_F [\Vector (A)]}_{\displaystyle J_x[\Vector (A)]} \Vector (dA)
\end {equation}
where $J_F (x) = - A^t A$ and where
\begin {equation}
\label {eqn:where}
J_F [\Vector (A)]
=
\left[
\setlength {\arraycolsep} {0.25em}
\begin {array} {c c c}
r^t\\
& \ddots\\
&& \hspace {0.5em} r^t
\end {array}
\right]
-
\left[
\setlength {\arraycolsep} {0.33em}
\begin {array} {c c c c c}
x_1 A^t& \cdots& x_n A^t
\end {array}
\right] ,
\end {equation}
in which $r = b - A x$ is the least-squares residual, and $x_i$ is the $i$-th entry of $x$.
\subsection {Transpose formula for condition numbers}
\label {sec:transpose}
The desired condition number is the norm induced from the norms in equation (\ref {eqn:norms}).
\begin {equation}
\label {eqn:induced}
\setlength {\arraycolsep} {0.25em}
\begin {array} {r c l}
\| \JxA \|
& =
& \displaystyle \max_{\DA} {\| \JxA \, \Vector (\DA) \|_\scaleX \over \| \Vector (\DA) \|_\scaleA}
\\ \noalign {
}
& =
& \displaystyle {\scaleA \over \scaleX} \max_{\DA} {\| \JxA \, \Vector (\DA) \|_2 \over \| \DA \|_2}
\end {array}
\end {equation}
The numerator and denominator are vector and matrix $2$-norms, respectively. If $A$ is an $m \times n$ matrix, then the maximization in equation (\ref {eqn:induced}) has many degrees of freedom. An identity for operator norms can be applied to avoid this large optimization problem.
Suppose $\reals^M$ and $\reals^N$ have the norms $\| \cdot \|_M$ and $\| \cdot \|_N$, respectively. If a problem with data $d \in \reals^N$ has a solution function $s = f(d) \in \reals^M$, then the condition number is the induced norm of the $M \times N$ Jacobian matrix,
\vspace {-1ex}
\begin {equation}
\label {eqn:condition}
\| J_f (d) \| = \max_{\Dd} {\| J_f(d) \, \Dd \|_M \over \| \Dd \|_N} \, .
\end {equation}
This optimization problem has $N$ degrees of freedom. An alternate expression is the norm for the transposed operator represented by the transposed matrix,\footnote {Equation (\ref {eqn:transpose}) is stated by
\citet [chp.\ IV, p.\ 7, eqn.\ 4] {BourbakiTVS},
\citet [p.\ 478, lem.\ 2] {DunfordSchwartz1958},
\citet [p.\ 93, thm.\ 4.10, eqn.\ 2] {Rudin1973},
and
\citet [p.\ 195, thm.\ 2$^\prime$, eqn.\ 3] {Yosida1974}.
The name of the transposed operator varies. See \citet [chp.\ IV, p.\ 6, top] {BourbakiTVS} for ``transpose,'' Dunford and Schwartz (loc.\ cit.)\ and Rudin (loc.\ cit.)\ for ``adjoint,'' and \citet [p.\ 194, def.\ 1] {Yosida1974} for ``conjugate'' or ``dual.'' Some parts of mathematics use ``adjoint'' in the restricted context of Hilbert spaces, for example in linear algebra see \citet [pp.\ 168--174, sec.\ 5.1] {Lancaster1985}. That concept is actually a ``slightly different notion'' \citep [p.\ 479] {DunfordSchwartz1958} from the Banach space transpose used here.
}
\begin {equation}
\label {eqn:transpose}
\| J_f (d) \| = \| [J_f (d)]^t \|^* = \max_{\Ds} {\| [J_f (d)]^t \Ds \|_N^* \over \| \Ds \|_M^*} \, .
\end {equation}
The norm is induced from the dual norms $\| \cdot \|_M^*$ and $\| \cdot \|_N^*$ which must be determined. This optimization problem has $M$ degrees of freedom. Equation (\ref {eqn:transpose}) might be easier to evaluate, especially when the problem has many more data values than solution variables, $N \gg M$, as is often the case.
Applying the formula (\ref {eqn:transpose}) for the norm of the transpose matrix to the equation (\ref {eqn:induced}) results in the simpler optimization problem,
\begin {equation}
\label {eqn:simpler}
\| \JxA \| =
{\scaleA \over \scaleX} \max_{\Dx} {\| \{ \JxA \}^t \Dx \|^*_2 \over \| \Dx \|^*_2}
\end {equation}
The norm for the transposed Jacobian matrix is induced from the duals of the $2$-norms for matrices and vectors. The vector $2$-norm in the denominator is its own dual. So as not to interrupt the present discussion, some facts needed to evaluate the numerator are proved in section \ref {sec:advanced}: the dual of the matrix $2$-norm is the nuclear norm (section \ref {app:spectral}), and a formula for the nuclear norm is given (section \ref {app:rank2}).
\subsection {Condition number of {\itshape x\/} with respect to {\itshape A\/}, continued}
\label {sec:conditionwrtAcontinued}
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\subsection {Transpose formula for condition numbers}
\label {sec:transpose}
The desired condition number is the norm induced from the norms in equation (\ref {eqn:norms}).
\begin {equation}
\label {eqn:induced}
\setlength {\arraycolsep} {0.25em}
\begin {array} {r c l}
\| \JxA \|
& =
& \displaystyle \max_{\DA} {\| \JxA \, \Vector (\DA) \|_\scaleX \over \| \Vector (\DA) \|_\scaleA}
\\ \noalign {
}
& =
& \displaystyle {\scaleA \over \scaleX} \max_{\DA} {\| \JxA \, \Vector (\DA) \|_2 \over \| \DA \|_2}
\end {array}
\end {equation}
The numerator and denominator are vector and matrix $2$-norms, respectively. If $A$ is an $m \times n$ matrix, then the maximization in equation (\ref {eqn:induced}) has many degrees of freedom. An identity for operator norms can be applied to avoid this large optimization problem.
Suppose $\reals^M$ and $\reals^N$ have the norms $\| \cdot \|_M$ and $\| \cdot \|_N$, respectively. If a problem with data $d \in \reals^N$ has a solution function $s = f(d) \in \reals^M$, then the condition number is the induced norm of the $M \times N$ Jacobian matrix,
\vspace {-1ex}
\begin {equation}
\label {eqn:condition}
\| J_f (d) \| = \max_{\Dd} {\| J_f(d) \, \Dd \|_M \over \| \Dd \|_N} \, .
\end {equation}
This optimization problem has $N$ degrees of freedom. An alternate expression is the norm for the transposed operator represented by the transposed matrix,\footnote {Equation (\ref {eqn:transpose}) is stated by
\citet [chp.\ IV, p.\ 7, eqn.\ 4] {BourbakiTVS},
\citet [p.\ 478, lem.\ 2] {DunfordSchwartz1958},
\citet [p.\ 93, thm.\ 4.10, eqn.\ 2] {Rudin1973},
and
\citet [p.\ 195, thm.\ 2$^\prime$, eqn.\ 3] {Yosida1974}.
The name of the transposed operator varies. See \citet [chp.\ IV, p.\ 6, top] {BourbakiTVS} for ``transpose,'' Dunford and Schwartz (loc.\ cit.)\ and Rudin (loc.\ cit.)\ for ``adjoint,'' and \citet [p.\ 194, def.\ 1] {Yosida1974} for ``conjugate'' or ``dual.'' Some parts of mathematics use ``adjoint'' in the restricted context of Hilbert spaces, for example in linear algebra see \citet [pp.\ 168--174, sec.\ 5.1] {Lancaster1985}. That concept is actually a ``slightly different notion'' \citep [p.\ 479] {DunfordSchwartz1958} from the Banach space transpose used here.
}
\begin {equation}
\label {eqn:transpose}
\| J_f (d) \| = \| [J_f (d)]^t \|^* = \max_{\Ds} {\| [J_f (d)]^t \Ds \|_N^* \over \| \Ds \|_M^*} \, .
\end {equation}
The norm is induced from the dual norms $\| \cdot \|_M^*$ and $\| \cdot \|_N^*$ which must be determined. This optimization problem has $M$ degrees of freedom. Equation (\ref {eqn:transpose}) might be easier to evaluate, especially when the problem has many more data values than solution variables, $N \gg M$, as is often the case.
Applying the formula (\ref {eqn:transpose}) for the norm of the transpose matrix to the equation (\ref {eqn:induced}) results in the simpler optimization problem,
\begin {equation}
\label {eqn:simpler}
\| \JxA \| =
{\scaleA \over \scaleX} \max_{\Dx} {\| \{ \JxA \}^t \Dx \|^*_2 \over \| \Dx \|^*_2}
\end {equation}
The norm for the transposed Jacobian matrix is induced from the duals of the $2$-norms for matrices and vectors. The vector $2$-norm in the denominator is its own dual. So as not to interrupt the present discussion, some facts needed to evaluate the numerator are proved in section \ref {sec:advanced}: the dual of the matrix $2$-norm is the nuclear norm (section \ref {app:spectral}), and a formula for the nuclear norm is given (section \ref {app:rank2}).
\subsection {Condition number of {\itshape x\/} with respect to {\itshape A\/}, continued}
\label {sec:conditionwrtAcontinued}
The application of equation (\ref {eqn:simpler}) requires evaluating the matrix-vector product in the numerator. Continuing the derivation of section \ref {sec:begin} from equation (\ref {eqn:where}), the vector-matrix product $v^t J_F [\Vector (A)]$ for some $v$ can be evaluated by straightforward multiplication,
\begin {displaymath}
v^t J_F [\Vector (A)]
=
\left[
\setlength {\arraycolsep} {0.25em}
\begin {array} {c c c}
v_1 r^t
& \cdots
& v_n r^t
\end {array}
\right]
-
\left[
\setlength {\arraycolsep} {0.33em}
\begin {array} {c c c c c}
x_1 v^t A^t & \cdots& x_n v^t A^t
\end {array}
\right] .
\end {displaymath}
This row vector, when transposed, is expressed more simply using $\Vector$ notation: the first part is $r$ scaled by each entry of $v$, $\Vector (r v^t)$, the second part is $A v$ scaled by each entry of $x$, $\Vector (A v x^t) $. Altogether
\begin {displaymath}
J_F [\Vector (A)]^t v
=
\Vector (rv^t - Avx^t) \, .
\end {displaymath}
Substituting $v = \{- [J_F (x)]^{-1}\}^t \Dx = (A^t A)^{-1} \Dx$ for some $\Dx$ gives, by equation (\ref {eqn:hence}),
\begin {displaymath}
\JxA^t \Dx
=
\Vector \left\{ r \, [(A^t A)^{-1} \Dx]^t - A \, [(A^t A)^{-1} \Dx] \, x^t \right\} \, ,
\end {displaymath}
or equivalently,
\begin {equation}
\label {eqn:JxADx}
\Matrix \{ \JxA^t \Dx \}
=
u_1^{} v_1^t + u_2^{} v_2^t
\end {equation}
where ``$\Matrix$'' is the inverse of ``$\Vector$,'' and
\begin {equation}
\label {eqn:vectors}
\setlength {\arraycolsep} {0.125em}
\begin {array} {r c l r c l}
u_1& =& r = b - Ax,&
v_1& =& (A^t A)^{-1} \Dx,\\ \noalign {
}
u_2& =& - A (A^t A)^{-1} \Dx, \qquad&
v_2& =& x .
\end {array}
\end {equation}
The matrix on the right side of equation (\ref {eqn:JxADx}) has rank $2$. Moreover, the two rank $1$ pieces are mutually orthogonal because the least-squares residual $r$ is orthogonal to the coefficient matrix $A$. With these replacements equation (\ref {eqn:simpler}) becomes
\begin {equation}
\label {eqn:inducedtransposed}
\big \| \JxA \big\|
=
{\scaleA \over \scaleX} \max_{\| \Dx \|_2 = 1} \| u_1^{} v_1^t + u_2^{} v_2^t \|^*_2 \, .
\end {equation}
Lemma \ref {lem:kahan} shows that the dual of the spectral matrix norm is the matrix norm that sums the singular values of the matrix, which is called the nuclear norm. Lemma \ref {lem:rank2} then evaluates this norm for rank $2$ matrices to find that the objective function of equation (\ref {eqn:inducedtransposed}) is
\begin {equation}
\label {eqn:numerator}
\sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2 \|_2 \, \cos (\theta_u - \theta_v)} \, ,
\end {equation}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and $\theta_v$ is the angle between $v_1$ and $v_2$, and both angles should be taken from $0$ to $\pi$. Since $u_1$ is orthogonal to $u_2$ therefore $\theta_u = \pi / 2$ and then $| \theta_u - \theta_v | \le \pi / 2$ so $\cos (\theta_u - \theta_v)$ is not negative. This means the maximum lies between the lower and upper limits
\begin {equation}
\label {eqn:lowerandupper}
\sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2} \quad \mbox {and} \quad \| u_1 \|_2 \, \| v_1 \|_2 + \| u_2 \|_2 \, \| v_2 \|_2 \, .
\end {equation}
With $\| \Dx \|_2$ restricted to $1$, the lower bound and also the upper bound attain their maxima when $\Dx$ is a right singular unit vector for the smallest singular value of $A$,
\begin {equation}
\label {eqn:maxima}
\| u_1 \|_2 \, \| v_1 \|_2 = {\| r \|_2 \over (\sigmamina)^2} \quad \mbox {and} \quad \| u_2 \|_2 \, \| v_2 \|_2 = {\| x \|_2 \over \sigmamina} \, .
\end {equation}
Some value of $\| u_1^{} v_1^t + u_2^{} v_2^t \|^*_2$ lies between the limits when $\Dx$ is a right singular unit vector for the smallest singular value of $A$. Because these are the largest possible limits, the maximum value must lie between them as well. These limits must be multiplied by the coefficient $\scaleA / \scaleX$ in equation (\ref {eqn:inducedtransposed}) to obtain bounds for the norm of the Jacobian matrix.
\subsection {Summary of condition numbers}
\label {sec:conditionwrtAfinished}
\ \par
\begin {theorem}
[\scshape Spectral condition numbers]
\label {thm:condition-numbers}
For the full column rank linear least-squares problem with solution $x = (A^tA)^{-1} A^t b$, and for the scaled norms of equation (\ref {eqn:norms}) with scale factors $\scaleA$, $\scaleB$, and $\scaleX$,
\begin {displaymath}
\chi_x (b) = {\scaleB \over \scaleX \sigmamin}
\qquad
\chi_x (A) = {\scaleA \over \scaleX} \max_{\| \Delta x \|_2 = 1} \sigma_1 + \sigma_2 \, ,
\end {displaymath}
where $\sigma_1$ and $\sigma_2$ are the singular values of the rank $2$ matrix $u_1^{} v_1^t + u_2^{} v_2^t$ \,for
\begin {displaymath}
\setlength {\arraycolsep} {0.125em}
\begin {array} {r c l r c l}
u_1& =& r = b - Ax,&
v_1& =& (A^t A)^{-1} \Dx,\\ \noalign {
}
u_2& =& - A (A^t A)^{-1} \Dx, \qquad&
v_2& =& x .
\end {array}
\end {displaymath}
The value of $\chi_x (A)$ lies between the lower limit of Malyshev and the upper limit of Bj\"orck,
\begin {displaymath}
{\scaleA \over \scaleX \sigmamin} \sqrt { \left( {\| r \|_2 \over \sigmamin} \right)^2 + \| x \|_2^2}
\, \le \,
\chi_x (A)
\, \le \,
{\scaleA \over \scaleX \sigmamin} \left( {\| r \|_2 \over \sigmamin} + \| x \|_2 \right) \, .
\end {displaymath}
The upper bound exceeds $\chi_x (A)$ by at most a factor $\sqrt 2$. The formula for $\chi_x(b)$ and the limits for $\chi_x(A)$ simplify to equations (\ref {eqn:chixA}, \ref {eqn:chixb}) for the scale factors in equation (\ref {eqn:scale-factors}).
\end {theorem}
\begin {proof}
Section \ref {sec:conditionwrtb} derives $\chi_x(b)$, and sections \ref {sec:begin}--\ref {sec:conditionwrtAfinished} derive $\chi_x(A)$ and the bounds.
\end {proof}
\section {Discussion}
\label {sec:comparison}
\subsection {Example of strict limits}
The condition number $\chi_x (A)$ in theorem \ref {thm:condition-numbers} can lie strictly between the limits of Bj\"orck and Malyshev. For the example of section \ref {sec:example}, the rank $2$ matrix in the theorem is
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
u_1^{} v_1^t + u_2^{} v_2^t = \left[ \begin {array} {c c}
- \beta \cos (\phi) \Delta x_1& - \beta \sin (\phi) \Delta x_1 / \alpha\\
- \beta \cos (\phi) \Delta x_2 / \alpha& - \beta \sin (\phi) \Delta x_2 / \alpha^2\\
\Delta x_1& \Delta x_2 / \alpha^2 \\ \end {array} \right] .
\end {displaymath}
For the specific values $\alpha = 1/10$, $\beta = 1$, $\phi = \pi / 10$, the sum of the singular values of this matrix can be numerically maximized over $\| \Delta x \|_2 = 1$ to evaluate the condition number with the following results.
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
\renewcommand {\tabcolsep} {0.25em}
\begin {tabular} {r c l l}
$\displaystyle \boldkappa (\vds \tan (\boldtheta) + 1)$& $=$& 40.928\ldots& upper limit of Bj\"orck\\
$\displaystyle \chi_x (A)$& $=$& 35.193\ldots& condition number\\
$\displaystyle \boldkappa \, \big( \strut [\vds \tan (\boldtheta)]^2 + 1\big)^{1/2}$& $=$& 32.505\ldots& lower limit of Malyshev\\
\end {tabular}
\end {displaymath}
These calculations were done with Mathematica \citep {Wolfram2003}.
\subsection {Exact formulas for some condition numbers}
\label {sec:exact}
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\subsection {Summary of condition numbers}
\label {sec:conditionwrtAfinished}
\ \par
\begin {theorem}
[\scshape Spectral condition numbers]
\label {thm:condition-numbers}
For the full column rank linear least-squares problem with solution $x = (A^tA)^{-1} A^t b$, and for the scaled norms of equation (\ref {eqn:norms}) with scale factors $\scaleA$, $\scaleB$, and $\scaleX$,
\begin {displaymath}
\chi_x (b) = {\scaleB \over \scaleX \sigmamin}
\qquad
\chi_x (A) = {\scaleA \over \scaleX} \max_{\| \Delta x \|_2 = 1} \sigma_1 + \sigma_2 \, ,
\end {displaymath}
where $\sigma_1$ and $\sigma_2$ are the singular values of the rank $2$ matrix $u_1^{} v_1^t + u_2^{} v_2^t$ \,for
\begin {displaymath}
\setlength {\arraycolsep} {0.125em}
\begin {array} {r c l r c l}
u_1& =& r = b - Ax,&
v_1& =& (A^t A)^{-1} \Dx,\\ \noalign {
}
u_2& =& - A (A^t A)^{-1} \Dx, \qquad&
v_2& =& x .
\end {array}
\end {displaymath}
The value of $\chi_x (A)$ lies between the lower limit of Malyshev and the upper limit of Bj\"orck,
\begin {displaymath}
{\scaleA \over \scaleX \sigmamin} \sqrt { \left( {\| r \|_2 \over \sigmamin} \right)^2 + \| x \|_2^2}
\, \le \,
\chi_x (A)
\, \le \,
{\scaleA \over \scaleX \sigmamin} \left( {\| r \|_2 \over \sigmamin} + \| x \|_2 \right) \, .
\end {displaymath}
The upper bound exceeds $\chi_x (A)$ by at most a factor $\sqrt 2$. The formula for $\chi_x(b)$ and the limits for $\chi_x(A)$ simplify to equations (\ref {eqn:chixA}, \ref {eqn:chixb}) for the scale factors in equation (\ref {eqn:scale-factors}).
\end {theorem}
\begin {proof}
Section \ref {sec:conditionwrtb} derives $\chi_x(b)$, and sections \ref {sec:begin}--\ref {sec:conditionwrtAfinished} derive $\chi_x(A)$ and the bounds.
\end {proof}
\section {Discussion}
\label {sec:comparison}
\subsection {Example of strict limits}
The condition number $\chi_x (A)$ in theorem \ref {thm:condition-numbers} can lie strictly between the limits of Bj\"orck and Malyshev. For the example of section \ref {sec:example}, the rank $2$ matrix in the theorem is
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
u_1^{} v_1^t + u_2^{} v_2^t = \left[ \begin {array} {c c}
- \beta \cos (\phi) \Delta x_1& - \beta \sin (\phi) \Delta x_1 / \alpha\\
- \beta \cos (\phi) \Delta x_2 / \alpha& - \beta \sin (\phi) \Delta x_2 / \alpha^2\\
\Delta x_1& \Delta x_2 / \alpha^2 \\ \end {array} \right] .
\end {displaymath}
For the specific values $\alpha = 1/10$, $\beta = 1$, $\phi = \pi / 10$, the sum of the singular values of this matrix can be numerically maximized over $\| \Delta x \|_2 = 1$ to evaluate the condition number with the following results.
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
\renewcommand {\tabcolsep} {0.25em}
\begin {tabular} {r c l l}
$\displaystyle \boldkappa (\vds \tan (\boldtheta) + 1)$& $=$& 40.928\ldots& upper limit of Bj\"orck\\
$\displaystyle \chi_x (A)$& $=$& 35.193\ldots& condition number\\
$\displaystyle \boldkappa \, \big( \strut [\vds \tan (\boldtheta)]^2 + 1\big)^{1/2}$& $=$& 32.505\ldots& lower limit of Malyshev\\
\end {tabular}
\end {displaymath}
These calculations were done with Mathematica \citep {Wolfram2003}.
\subsection {Exact formulas for some condition numbers}
\label {sec:exact}
\begin {table}
\caption {\it Cases for which condition numbers have been determined for the full column rank least-squares problem, $\min_x \| b - A x \|_2$. All the formulas are for $\chi_x(A)$ except Grattan's formula is for $\chi_x(A,b)$. Notation: $r$ is the residual, $\sigmamin$ is the smallest singular value of $A$. In column 5, ``approx'' means the value in column 4 approximates the condition number, ``exact'' means it is the condition number for the chosen norms.}
\label {tab:LLS}
\newcommand {\textstrut} {\vrule depth1.125ex height2.375ex width\strutwidth}
\newcommand {\leftbox} [3] {\begin {minipage}{#1} \vrule depth0ex height4.0ex width\strutwidth #2\\ \scriptsize #3\vrule depth3.0ex height0ex width\strutwidth \end {minipage}}
\setlength {\arraycolsep} {0.6em}
\vspace {-4ex}
\begin {displaymath}
\small
\begin {array} {| l | c | c | c | c |}
\cline {2-3}
\multicolumn {1} { c |} {\textstrut}
& \multicolumn {2} { c |} {\mbox {norms}}
\\
\hline
\multicolumn {1} {| c |} {\textstrut \mbox {source}}
& \mbox {data}
& \mbox {solution}
& \mbox {formula}
& \mbox {status}
\\ \hline \hline
\leftbox {10em}
{Bj\"orck and theorem \ref {thm:condition-numbers}}
{\nocite {Bjorck1996}(1996, p.\ 31, eqn.\ 1.4.28)}
& \displaystyle {\| \DA \|_2 \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \displaystyle {\| A \|_2 \over \strut \sigmamin} \left( {\| r \|_2 \over \strut \sigmamin \, \| x \|_2} \, + 1 \right)
& \mbox {approx}
\\
\leftbox {10em}
{Geurts}
{\nocite {Geurts1982}(1982, p.\ 93, eqn.\ 4.3)}
& \displaystyle {\| \DA \|\sub{F} \over \| A \|\sub{F}}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \displaystyle {\| A \|\sub{F} \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \sigmamin^2 \, \| x \|_2^2} + 1}
& \mbox {exact}
\\
\multicolumn {1} {| l} {\leftbox {10em} {Gratton} {\nocite {Gratton1996}(1996, p.\ 525, eqn.\ 2.1)}}
& \hspace {-4em} \displaystyle \left\| \strut [ \alpha \, \DA, \beta \, \Db \, ] \right \|\sub{F}
& \| \Dx \|_2
& \displaystyle {1 \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \alpha^2 \sigmamin^2} + {\| x \|_2^2 \over \alpha^2} + {1 \over \beta^2}}
& \mbox {exact}
\\
\leftbox {10em}
{Malyshev}
{\nocite {Malyshev2003}(2003, p.\ 1187, eqn.\ 1.8)}
& \displaystyle {\| \DA \|\sub{F} \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
&& \mbox {exact}
\\
\leftbox {11.5em}
{Malyshev and theorem \ref {thm:condition-numbers}}
{\nocite {Malyshev2003}(2003, p.\ 1189, eqn.\ 2.4 and line --6)}
& \displaystyle {\| \DA \|_2 \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \mbox{\hspace {-1.5em} \raisebox{4.0ex}[0pt][0pt]{$\displaystyle \left.
\vrule depth7ex height0ex width 0pt \right\} \hspace {0.5em} {\| A \|_2 \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \sigmamin^2 \, \| x \|_2^2} +
1}$}}
& \mbox {approx}
\\ \hline
\end {array}
\end {displaymath}
\vspace*{-2ex}
\end {table}
Table \ref {tab:LLS} lists several condition numbers or approximations to condition numbers for least-squares solutions. The three exact values measure changes to $A$ by the Frobenius norm, while the two approximate values are for the spectral norm. The difference can be attributed to the ease or difficulty of solving the maximization problem in equation (\ref {eqn:inducedtransposed}). The dual spectral norm of the rank $2$ matrix involves a trigonometric function, $\cos (\pi / 2 - \theta_v)$ in equation (\ref {eqn:numerator}), whose value only can be estimated. If a Frobenius norm were used instead, then lemma \ref {lem:rank2} shows the dual norm of the rank $2$ matrix involves an expression, $\cos (\pi / 2) \cos (\theta_v)$, whose value is zero, which greatly simplifies the maximization problem.
\subsection {Overestimates of condition numbers}
\label {sec:overestimate}
Many error bounds in the literature combine $\chi_x(A) + \chi_x(b)$ in the manner of equation (\ref {eqn:double}),
\begin {eqnarray}
\nonumber
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \chi_x (A,b) \, \epsilon + o (\epsilon) \qquad \mbox {attainable}
\\
\label {eqn:bestbound-1}
& \le
& \big[ \chi_x(A) + \chi_x(b) \big] \epsilon + o (\epsilon) \qquad \mbox {overestimate by at most $\times 2$}
\\ \noalign {
}
\label {eqn:bestbound-2}
& \le
& \big[ \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta) \big] \epsilon + o (\epsilon) \quad \mbox {further at most $\times \sqrt 2$}
\\ \noalign {
}
\label {eqn:bestbound-3}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| b \|_2 \over \sigmamin \| x \|_2} \right) \epsilon +o (\epsilon) \, ,
\end {eqnarray}
where $\epsilon = \max \{ \| \DA \|_2 / \| A \|_2, \| \Db \|_2 / \| b \|_2 \}$. Bounds (\ref {eqn:bestbound-1}, \ref {eqn:bestbound-2}) are larger than the attainable bound by at most factors $2$ and $2 \sqrt 2$, respectively, by equation (\ref {eqn:sum-6}) and theorem \ref {thm:condition-numbers}.
Some bounds are yet larger. \citet [p.\ 382, eqn.\ 20.1] {Higham2002} reports
\begin {eqnarray*}
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \boldkappa \epsilon \left( 2 + ( \boldkappa + 1) {\| r \|_2 \over \| A \|_2 \| x \|_2} \right) + {\mathcal O} (\epsilon^2)
\\ \noalign {
}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| A \|_2 \| x \|_2 + \| r \|_2 \over \sigmamin \| x \|_2} \right) \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {eqnarray*}
This bound is an overestimate in comparison to equation (\ref {eqn:bestbound-3}).
An egregious overestimate occurs in an error bound that appears to have originated in the 1983 edition of the popular textbook of \citet [p.\ 242, eqn.\ 5.3.8] {Golub1996}. The overestimate is restated by \citet [p.\ 50] {Anderson1992} in the LAPACK documentation, and by \citet [p.\ 117] {Demmel1997},
\begin {equation}
\label {eqn:GVL}
{\| \Dx \|_2 \over \| x \|_2}
\le
\left[ 2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \right] \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {equation}
In comparison with equation (\ref {eqn:bestbound-2}) this bound replaces $\vds$ by $\boldkappa$ and replaces $1$ by $\sec (\boldtheta)$. An overestimate by a factor of $\boldkappa$ occurs for the example of section \ref {sec:example} with $\alpha \ll 1$, $\beta = 1$, and $\phi = {\pi \over 2}$. In this case the ratio of equation (\ref {eqn:GVL}) to equation (\ref {eqn:bestbound-2}) is
\begin {displaymath}
{2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \over \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta)} = {\displaystyle {1 + 2 \alpha \sqrt {1 + \beta^2} \over \alpha^2 \beta} \over \displaystyle {1 + \beta + \alpha \sqrt {1 + \beta^2} \over \alpha \beta}} \approx {1 \over 2 \, \alpha} = {\boldkappa \over 2} \, .
\end {displaymath}
\section {Norms of operators on normed linear spaces of finite dimension}
\label {sec:advanced}
\subsection {Introduction}
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Table \ref {tab:LLS} lists several condition numbers or approximations to condition numbers for least-squares solutions. The three exact values measure changes to $A$ by the Frobenius norm, while the two approximate values are for the spectral norm. The difference can be attributed to the ease or difficulty of solving the maximization problem in equation (\ref {eqn:inducedtransposed}). The dual spectral norm of the rank $2$ matrix involves a trigonometric function, $\cos (\pi / 2 - \theta_v)$ in equation (\ref {eqn:numerator}), whose value only can be estimated. If a Frobenius norm were used instead, then lemma \ref {lem:rank2} shows the dual norm of the rank $2$ matrix involves an expression, $\cos (\pi / 2) \cos (\theta_v)$, whose value is zero, which greatly simplifies the maximization problem.
\subsection {Overestimates of condition numbers}
\label {sec:overestimate}
Many error bounds in the literature combine $\chi_x(A) + \chi_x(b)$ in the manner of equation (\ref {eqn:double}),
\begin {eqnarray}
\nonumber
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \chi_x (A,b) \, \epsilon + o (\epsilon) \qquad \mbox {attainable}
\\
\label {eqn:bestbound-1}
& \le
& \big[ \chi_x(A) + \chi_x(b) \big] \epsilon + o (\epsilon) \qquad \mbox {overestimate by at most $\times 2$}
\\ \noalign {
}
\label {eqn:bestbound-2}
& \le
& \big[ \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta) \big] \epsilon + o (\epsilon) \quad \mbox {further at most $\times \sqrt 2$}
\\ \noalign {
}
\label {eqn:bestbound-3}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| b \|_2 \over \sigmamin \| x \|_2} \right) \epsilon +o (\epsilon) \, ,
\end {eqnarray}
where $\epsilon = \max \{ \| \DA \|_2 / \| A \|_2, \| \Db \|_2 / \| b \|_2 \}$. Bounds (\ref {eqn:bestbound-1}, \ref {eqn:bestbound-2}) are larger than the attainable bound by at most factors $2$ and $2 \sqrt 2$, respectively, by equation (\ref {eqn:sum-6}) and theorem \ref {thm:condition-numbers}.
Some bounds are yet larger. \citet [p.\ 382, eqn.\ 20.1] {Higham2002} reports
\begin {eqnarray*}
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \boldkappa \epsilon \left( 2 + ( \boldkappa + 1) {\| r \|_2 \over \| A \|_2 \| x \|_2} \right) + {\mathcal O} (\epsilon^2)
\\ \noalign {
}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| A \|_2 \| x \|_2 + \| r \|_2 \over \sigmamin \| x \|_2} \right) \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {eqnarray*}
This bound is an overestimate in comparison to equation (\ref {eqn:bestbound-3}).
An egregious overestimate occurs in an error bound that appears to have originated in the 1983 edition of the popular textbook of \citet [p.\ 242, eqn.\ 5.3.8] {Golub1996}. The overestimate is restated by \citet [p.\ 50] {Anderson1992} in the LAPACK documentation, and by \citet [p.\ 117] {Demmel1997},
\begin {equation}
\label {eqn:GVL}
{\| \Dx \|_2 \over \| x \|_2}
\le
\left[ 2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \right] \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {equation}
In comparison with equation (\ref {eqn:bestbound-2}) this bound replaces $\vds$ by $\boldkappa$ and replaces $1$ by $\sec (\boldtheta)$. An overestimate by a factor of $\boldkappa$ occurs for the example of section \ref {sec:example} with $\alpha \ll 1$, $\beta = 1$, and $\phi = {\pi \over 2}$. In this case the ratio of equation (\ref {eqn:GVL}) to equation (\ref {eqn:bestbound-2}) is
\begin {displaymath}
{2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \over \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta)} = {\displaystyle {1 + 2 \alpha \sqrt {1 + \beta^2} \over \alpha^2 \beta} \over \displaystyle {1 + \beta + \alpha \sqrt {1 + \beta^2} \over \alpha \beta}} \approx {1 \over 2 \, \alpha} = {\boldkappa \over 2} \, .
\end {displaymath}
\section {Norms of operators on normed linear spaces of finite dimension}
\label {sec:advanced}
\subsection {Introduction}
This section describes the dual norms in the formulas of sections \ref {sec:transpose} and \ref {sec:conditionwrtAcontinued}. The actual mathematical concept is a norm for the dual space. However, linear algebra ``identifies'' a space with its dual, so the concept becomes a ``dual norm'' for the same space. This point of view is appropriate for Hilbert spaces, but it omits an important level of abstraction. As a result, the linear algebra literature lacks a complete development of finite dimensional normed linear (Banach) spaces. Rather than make functional analysis a prerequisite for this paper, here the identification approach is generalized to give dual norms for spaces other than column vectors (which is needed for data in matrix form), but only as far as the dual norm itself in section \ref {app:duals}. Section \ref {app:spectral} gives the formula for the dual of the spectral matrix norm. Section \ref {app:rank2} evaluates the norm for matrices of rank $2$.
{\it Banach spaces are needed in this paper because the norms used in numerical analysis are not necessarily those of a Hilbert space.} The space of $m \times n$ matrices viewed as column vectors has been given the spectral matrix norm in equation (\ref {eqn:norms}). If the norm were to make the space a Hilbert space, then the norm would be given by an inner product. There would be an $mn \times mn$ symmetric matrix, $S$, so that for every $m \times n$ matrix $B$,
\begin {displaymath}
\| B \|_2
=
\sqrt {\strut [\Vector (B)]^t \; S \; \Vector (B)} \, ,
\end {displaymath}
which is impossible.
\subsection {Duals of normed spaces}
\label {app:duals}
If $\X$ is a finite dimensional vector space over $\reals$, then the dual space $\X^*$ consists of all linear transformations $f : \X \rightarrow \reals$, called functionals. If $\X$ has a norm, then $\X^*$ has the usual operator norm given by
\begin {equation}
\label {eqn:operatorNorm}
\| f \| = \sup_{\| x \| = 1} f (x) \, .
\end {equation}
One notation is used for both norms because whether a norm is for $\X$ or $\X^*$ can be decided by what is inside.
For a finite dimensional $\X$ with a basis $e_1$, $e_2$, $\ldots\,$, $e_n$, the dual space has a basis $g_1$, $g_2$, $\ldots\,$, $f_n$ defined by $f_i (e_j) = \delta_{i,j}$ where $\delta_{i,j}$ is Kronecker's delta function. In linear algebra for finite dimensional spaces, it is customary to represent the arithmetic of $\X^*$ in terms of $\X$ under the transformation $T : \X \rightarrow \X^*$ defined on the bases by $T(e_i) = f_i$. \textit {This transformation is not unique because it depends on the choices of bases.\/} Usually $\X$ has a favored or ``canonical'' basis whose $T$ is said to ``identify'' $\X^*$ with $\X$. Under this identification the norm for the dual space then is regarded as a norm for the original space.
\begin {definition}
[\scshape Dual norm]
Let $\X = \reals^m$ have norm $\| \cdot \|$ and let $T$ identify $\X$ with the dual space $\X^*$. The dual norm for $\X$ is
\begin {displaymath}
\| v \|^* = \| T(v) \|
\end {displaymath}
where the right side is the norm in equation (\ref {eqn:operatorNorm}) for the dual space.
\end {definition}
The notation $\| \cdot \|^*$ avoids confusing the two norms for $\X$. There seems to be no standard notation for the dual norm; others are $\| \cdot \|_D$, $\| \cdot \|^D$, and $\| \cdot \|_{\rm d}$ which are used respectively by \citet [p.\ 107, eqn.\ 6.2] {Higham2002}, \citet [p.\ 275, def.\ 5.4.12] {Horn1985}, and \citet [p.\ 381, eqn.\ 1] {Lancaster1985}.
\subsection {Dual of the spectral matrix norm}
\label {app:spectral}
The space $\reals^{m \times n}$ of real $m \times n$ matrices has a canonical basis consisting of the matrices $E^{(i,j)}$ whose entries are zero except the $i,j$ entry which is $1$. This basis identifies a matrix $A$ with the functional whose value at a matrix $B$ is $\sum_{i,j} A_{i,j} B_{i,j} = \tr (A^t B)$.
\begin {lemma}
[\scshape Dual of the spectral matrix norm]
\label {lem:kahan}
The dual norm of the spectral matrix norm with respect to the aforementioned canonical basis for $\reals^{m \times n}$ is given by $\| A \|_2^* = \| \sigma(A) \|_1$, where $\sigma (A) \in \reals^{\min \{m, n\}}$ is the vector of $A$'s singular values including multiplicities. That is, $\| A \|_2^*$ is the sum of the singular values of $A$ with multiplicities, which is called the nuclear norm or the trace norm.
\end{lemma}
\begin {proof}
{\it (Supplied by \citet {Kahan2003}.)} Let $A = U \Sigma V^t$ be a ``full'' singular value decomposition of $A$, where both $U$ and $V$ are orthogonal matrices, and where $\Sigma$ is an $m \times n$ ``diagonal'' matrix whose diagonal entries are those of $\sigma (A)$. The trace of a square matrix, $M$, is invariant under conjugation, $V^{-1} M V$, so
\begin {displaymath}
\| A \|_2^*
=
\sup_{\| B \|_2 = 1} \tr (A^t B)
=
\sup_{\| B \|_2 = 1} \tr (V \Sigma^t \, U^t B)
=
\sup_{\| B \|_2 = 1} \tr (\Sigma^t \, U^t B \, V) \, .
\end {displaymath}
Since $\| U^t B V \|_2 = \| B \|_2 = 1$, the entries of $U^t B V$ are at most $1$ in magnitude, and therefore $| \tr (\Sigma^t \, U^t B \, V) | \le \tr (\Sigma^t)$. This upper bound is attained for $B = U D V^t$ where $D$ is the $m \times n$ ``identity'' matrix. \end {proof}
An alternate proof is offered by the work of \nocite {Taub1963} von \citet {vonNeumann1937siam}. He studied a special class of norms for $\reals^{m \times n}$. A symmetric gauge function of order $p$ is a norm for $\reals^p$ that is unchanged by every permutation and sign change of the entries of the vectors. Such a function applied to the singular values of a matrix always defines a norm on $\reals^{m \times n}$. For example, $\| A \|_2 = \| \sigma (A) \|_\infty$ where as in lemma \ref {lem:kahan} $\sigma (A)$ is the length $\min \{ m, n \}$ column vector of singular values for $A$. The dual of this norm is given by the dual norm for the singular values vector, see \citet [p.\ 78, lem.\ 3.5] {Stewart1990}.
\begin {proof}
{\it (In the manner of John von Neumann.)} By the aforementioned lemma to von Neumann's gauge theorem, $\| A \|_2^* = \| \sigma (A) \|_\infty^* = \| \sigma (A) \|_1$.
\end {proof}
\subsection {Rank 2 Matrices}
\label {app:rank2}
This section finds singular values of rank $2$ matrices to establish some norms of the matrices that simplify the condition numbers in equation (\ref {eqn:inducedtransposed}).
\begin {lemma}
[\scshape Frobenius and nuclear norms of rank 2 matrices.]
\label {lem:rank2}
If $u_1, u_2 \in \reals^m$ and $v_1, v_2 \in \reals^n$, then (Frobenius norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F}^* =
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F} =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u) \cos (\theta_v)} \, ,
\end {array}
\end {displaymath}
and (nuclear norm, or trace norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|_2^* =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u - \theta_v)} \, ,
\end {array}
\end {displaymath}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and $\theta_v$ is
the angle between $v_1$ and $v_2$. Both angles should be taken from $0$ to $\pi$.
\end{lemma}
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\subsection {Dual of the spectral matrix norm}
\label {app:spectral}
The space $\reals^{m \times n}$ of real $m \times n$ matrices has a canonical basis consisting of the matrices $E^{(i,j)}$ whose entries are zero except the $i,j$ entry which is $1$. This basis identifies a matrix $A$ with the functional whose value at a matrix $B$ is $\sum_{i,j} A_{i,j} B_{i,j} = \tr (A^t B)$.
\begin {lemma}
[\scshape Dual of the spectral matrix norm]
\label {lem:kahan}
The dual norm of the spectral matrix norm with respect to the aforementioned canonical basis for $\reals^{m \times n}$ is given by $\| A \|_2^* = \| \sigma(A) \|_1$, where $\sigma (A) \in \reals^{\min \{m, n\}}$ is the vector of $A$'s singular values including multiplicities. That is, $\| A \|_2^*$ is the sum of the singular values of $A$ with multiplicities, which is called the nuclear norm or the trace norm.
\end{lemma}
\begin {proof}
{\it (Supplied by \citet {Kahan2003}.)} Let $A = U \Sigma V^t$ be a ``full'' singular value decomposition of $A$, where both $U$ and $V$ are orthogonal matrices, and where $\Sigma$ is an $m \times n$ ``diagonal'' matrix whose diagonal entries are those of $\sigma (A)$. The trace of a square matrix, $M$, is invariant under conjugation, $V^{-1} M V$, so
\begin {displaymath}
\| A \|_2^*
=
\sup_{\| B \|_2 = 1} \tr (A^t B)
=
\sup_{\| B \|_2 = 1} \tr (V \Sigma^t \, U^t B)
=
\sup_{\| B \|_2 = 1} \tr (\Sigma^t \, U^t B \, V) \, .
\end {displaymath}
Since $\| U^t B V \|_2 = \| B \|_2 = 1$, the entries of $U^t B V$ are at most $1$ in magnitude, and therefore $| \tr (\Sigma^t \, U^t B \, V) | \le \tr (\Sigma^t)$. This upper bound is attained for $B = U D V^t$ where $D$ is the $m \times n$ ``identity'' matrix. \end {proof}
An alternate proof is offered by the work of \nocite {Taub1963} von \citet {vonNeumann1937siam}. He studied a special class of norms for $\reals^{m \times n}$. A symmetric gauge function of order $p$ is a norm for $\reals^p$ that is unchanged by every permutation and sign change of the entries of the vectors. Such a function applied to the singular values of a matrix always defines a norm on $\reals^{m \times n}$. For example, $\| A \|_2 = \| \sigma (A) \|_\infty$ where as in lemma \ref {lem:kahan} $\sigma (A)$ is the length $\min \{ m, n \}$ column vector of singular values for $A$. The dual of this norm is given by the dual norm for the singular values vector, see \citet [p.\ 78, lem.\ 3.5] {Stewart1990}.
\begin {proof}
{\it (In the manner of John von Neumann.)} By the aforementioned lemma to von Neumann's gauge theorem, $\| A \|_2^* = \| \sigma (A) \|_\infty^* = \| \sigma (A) \|_1$.
\end {proof}
\subsection {Rank 2 Matrices}
\label {app:rank2}
This section finds singular values of rank $2$ matrices to establish some norms of the matrices that simplify the condition numbers in equation (\ref {eqn:inducedtransposed}).
\begin {lemma}
[\scshape Frobenius and nuclear norms of rank 2 matrices.]
\label {lem:rank2}
If $u_1, u_2 \in \reals^m$ and $v_1, v_2 \in \reals^n$, then (Frobenius norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F}^* =
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F} =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u) \cos (\theta_v)} \, ,
\end {array}
\end {displaymath}
and (nuclear norm, or trace norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|_2^* =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u - \theta_v)} \, ,
\end {array}
\end {displaymath}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and $\theta_v$ is
the angle between $v_1$ and $v_2$. Both angles should be taken from $0$ to $\pi$.
\end{lemma}
\begin {proof}
If any of the vectors vanish, then the formulas are clearly true, so it may be assumed that the vectors are nonzero. The strategy of the proof is to represent the rank $2$ matrix as a $2 \times 2$ matrix whose singular values can be calculated. Since singular values are wanted, it is necessary that the bases for the $2 \times 2$ representation be orthonormal.
To that end, let $w_1$ and $w_2$ be orthogonal unit vectors with $u_1 = \alpha_1 w_1$ and $u_2 = \alpha_2 w_1 + \beta w_2$. The coefficients' signs are indeterminate, so without loss of generality assume $\alpha_1 \ge 0$ and $\beta \ge 0$, in which case
\begin {displaymath}
\alpha_1 = \| u_1 \|_2
\qquad
\alpha_2 = {u_1^t u_2^{} \over \| u_1 \|_2}
\qquad
\beta = \left\| u_2 - \left( {u_1^t u_2^{} \over \| u_1 \|_2} \right)
\left( {u_1 \over \| u_1 \|_2} \right) \, \right\|_2 \, .
\end {displaymath} Similarly, let $x_1$ and $x_2$ be mutually orthogonal
unit vectors with $v_1 = \gamma_1 x_1$ and $v_2 = \gamma_2 x_1 + \delta
x_2$. Again without loss of generality $\gamma_1 \ge 0$ and $\delta \ge 0$
so that
\begin {displaymath}
\gamma_1 = \| v_1 \|_2
\qquad
\gamma_2 = {v_1^t v_2^{} \over \| v_1 \|_2}
\qquad
\delta = \left\| v_2 - \left( {v_1^t v_2^{} \over \| v_1 \|_2} \right)
\left( {v_1 \over \| v_1 \|_2} \right) \, \right\|_2 \, .
\end {displaymath}
Notice that
\begin {displaymath}
\beta^2 = \| u_2 \|_2^2 - \left( {u_1^t u_2^{} \over \| u_1 \|_2} \right)^2
\qquad
\delta^2 = \| v_2 \|_2^2 - \left( {v_1^t v_2^{} \over \| v_1 \|_2}
\right)^2 \, .
\end {displaymath}
Let $G = u_1^{} v_1^t + u_2^{} v_2^t$. A straightforward calculation shows that, with respect to the orthonormal basis consisting of $x_1$ and $x_2$, the matrix $G^t G$ is represented by the matrix
\begin {displaymath}
M = \left[ \begin {array} {c c} {\beta }^2\,{{{\gamma }_2}}^2 +
{\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)
}^2& {\beta }^2\,\delta \,{{\gamma }_2} +
\delta \,{{\alpha }_2}\,
\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)\\ \noalign {
} {\beta
}^2\,\delta \,{{\gamma }_2} +
\delta \,{{\alpha }_2}\,
\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)& {\beta }^2\,{\delta }^2 +
{\delta }^2\,{{{\alpha }_2}}^2 \end {array} \right] \, .
\end {displaymath}
The desired norms are now given in terms of the eigenvalues of $M$, $\lambda_\pm$,
\begin {displaymath}
\| G \|\sub{F}
=
\sqrt {\lambda_+ + \lambda_-} = \sqrt {\tr (M)}
\quad \mbox {and} \quad
\| G \|_2^*
=
\sqrt {\lambda_+} + \sqrt {\lambda_-} \, .
\end {displaymath}
The expression for $\| G \|_2^*$ requires further analysis. For any $2 \times 2$ matrix $M$,
\begin {displaymath}
\lambda_\pm = {\tr (M) \over 2} \pm \sqrt { \left( {\tr (M) \over 2}
\right)^2 - \det (M)} \, .
\end {displaymath}
In the present case these eigenvalues are real because the $M$ of interest is symmetric, and $\det (M) \ge 0$ because it is also positive semidefinite. Altogether $[\tr (M)]^2 \ge 4 \det (M) \ge 0$, so $\tr (M) \ge 2 \det (M) \ge 0$. These bounds prove the following quantities are real, and it can be verified they are the square roots of the eigenvalues of $M$,
\begin {displaymath}
\sqrt {\lambda_\pm} = \sqrt {{\tr (M) \over 4} + \sqrt {\det (M) \over 4}}
\pm \sqrt {{\tr (M) \over 4} - \sqrt {\det (M) \over 4}} \, ,
\end {displaymath}
thus
\begin {displaymath}
\| G \|_2^*
=
\sqrt {\lambda_+} + \sqrt {\lambda_-}
=
\sqrt {\tr (M) + 2 \sqrt {\det (M)}} \, .
\end {displaymath}
In summary, the desired quantities $\| G \|_2^*$ and $\| G \|\sub{F}^{}$ have been expressed in terms of $\det (M)$ and $\tr (M)$ which the expression for $M$ expands into formulas of $\alpha_i$, $\beta$, $\gamma_i$, and $\delta$. These in turn expand to expressions of $u_i$ and $v_i$. It is remarkable that the ultimate expressions in terms of $u_i$ and $v_i$ are straightforward,
\begin {eqnarray*}
\tr (M)
& =
& \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \,
(u_1^t u_2^{}) (v_1^t v_2^{})\\
\noalign {
}
& =
& \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \, \|
u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2 \|_2 \, \cos (\theta_u)
\cos (\theta_v)\\
\noalign {
}
\det (M)
& =
& \left( \| u_1 \|_2^2 \, \| u_2 \|_2^2 - (u_1^t u_2^{})^2 \right) \left(
\| v_1 \|_2^2 \, \| v_2 \|_2^2 - (v_1^t v_2^{})^2 \right)\\
\noalign {
}
& =
& \left( \| u_1 \|_2^2 \, \| u_2 \|_2^2 (\sin (\theta_u))^2 \right) \left(
\| v_1 \|_2^2 \, \| v_2 \|_2^2 (\sin (\theta_v))^2 \right) \, ,
\end {eqnarray*}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and similarly for $\theta_v$. The formula for $\| G \|\sub{F}$ is established. The formula for $\| G \|_2^*$ simplifies, using the difference formula for cosine, to the one in the statement of the lemma. Since the positive root of $\sqrt {\det (M)}$ is wanted, the angles should be chosen from $0$ to $\pi$ so the squares of the sines can be removed without inserting a change of sign. These calculations have been verified with Mathematica \citep {Wolfram2003}.
\end {proof}
\raggedright
\bibliographystyle {plainnat}
\bibliographystyle {siam}
\end{document}
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\begin{document}
\title{Jumps, folds and hypercomplex structures}
\author{Roger Bielawski* \and Carolin Peternell}
\operatorname{ad}dress{Institut f\"ur Differentialgeometrie,
Leibniz Universit\"at Hannover,
Welfengarten 1, 30167 Hannover, Germany}
\email{[email protected]}
\thanks{Both authors are members of, and the second author is fully supported by the DFG Priority
Programme 2026 ``Geometry at infinity".}
\begin{abstract}
We investigate the geometry of the Kodaira moduli space $M$ of sections of $\pi:Z\to {\mathbb{P}}^1$, the normal bundle of which is allowed to jump from ${\mathcal{O}}(1)^{n}$ to ${\mathcal{O}}(1)^{n-2m}\oplus{\mathcal{O}}(2)^{m}\oplus{\mathcal{O}}^{m}$. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of $M$ extends to a logarithmic connection on $M$.
\end{abstract}
{\mathfrak s \mathfrak u}bjclass{53C26, 53C28}
\maketitle
\thispagestyle{empty}
\section{Introduction}
It is well known that a hyperk\"ahler or a hypercomplex structure on a smooth manifold $M$ can be encoded in the {\em twistor space}, which is a complex manifold $Z$ fibring over ${\mathbb{P}}^1$ and equipped with an antiholomorphic involution $\sigma$ covering the antipodal map. The manifold $M$ is recovered as the parameter space of $\sigma$-invariant sections with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$ ($n=\dim_{\mathbb C} M$). If we start with an arbitrary complex manifold $Z$ equipped with a holomorphic submersion $\pi:Z\to {\mathbb{P}}^1$ and an involution $\sigma$, then the corresponding component of the Kodaira moduli space of sections of $\pi$ will typically also contain sections with other normal bundles $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$. This kind of jumping normal bundle attracted recently attention in the case of $4$-dimensional hyperk\"ahler manifolds \cite{Hit2,Biq,Dun}, in the context of a phenomenon known as {\em folding} (one speaks then of {\em folded hyperk\"ahler metrics}).
\par
Folded hyperk\"ahler structures do not exhaust all geometric possibilities which arise when the normal bundle is allowed to jump. Even in four dimensions there are examples which are not folded (Example \operatorname{Re}f{proj} below). The aim of this paper is to investigate the natural extension of the hypercomplex geometry arising on such manifolds of sections (folded or not). More precisely, we are interested in the differential geometry of the (smooth) parameter space $ M$ of sections of $\pi:Z\to {\mathbb{P}}^1$ with normal bundle $N$ isomorphic $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$, where each $k_i{\mathfrak g}eq 0$. We shall discuss only the purely holomorphic case, i.e. we are interested in all sections, not just $\sigma$-invariant. Choosing an appropriate $\sigma$ allows one to carry over all results to hypercomplex or split hypercomplex manifolds.
\par
Our particular object of interest is the (holomorphic) {\em Obata connection} $\nabla$, i.e. the unique torsion free connection preserving the hypercomplex (or, rather, the biquaternionic, i.e. complexified hypercomplex) structure. This is defined on the open subset $U$ of $M$ corresponding to the sections with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$. The general twistor machinery (see, e.g. \cite{BE}) implies that $\nabla$ extends to a first order differential operator $D$ on sections of certain vector bundle defined over all of $M$.
Our point of view is to regard $D$ as a particular type of meromorphic connection with polar set ${\partial}elta=M\backslash U$. In general, this {\em meromorphic Obata connection} can have higher order poles along ${\partial}elta$. We show, however, that in the case when $M$ arises from a (partial) compactification of the twistor space of a hypercomplex manifold, the meromorphic Obata connection has a simple pole, and in fact it is then a logarithmic connection.
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\section{Geometry of jumps}
{\mathfrak s \mathfrak u}bsection{Two examples}
We begin with two examples illustrating the different geometric possibilities occuring when the normal bundle of a twistor line jumps. The first example is the basic example of a {\em hyperk\"ahler fold}, as explained by Hitchin \cite{Hit2}.
\begin{example} (Calabi-Eguchi-Hanson gravitational instanton) The twistor space $Z$ of the Calabi-Eguchi-Hanson metric is the resolution of the variety
$$\{(x,y,z)\in {\mathcal{O}}(2)\oplus{\mathcal{O}}(2)\oplus{\mathcal{O}}(2); xy=(z-p_1)(z-p_2)\},$$
where $p_i(\zeta)=a_i\zeta^2+2b_i\zeta+c_i$, $i=1,2$, are two fixed sections of ${\mathcal{O}}(2)$. The sections of the projection $Z\to {\mathbb{P}}^1$ can be described as follows \cite{Hit1}: let $z=a\zeta^2+2b\zeta+c$ be a section of ${\mathcal{O}}(2)$ and let $\alpha_i,\beta_i$ be the roots of $z-p_i$, $i=1,2$. Then the sections are given by
$$ z=a\zeta^2+2b\zeta+c, \enskip x=A(\zeta-\alpha_1)(\zeta-\alpha_2),\enskip y=B(\zeta-\beta_1)(\zeta-\beta_2),$$
where $AB=(a-a_1)(a-a_2)$.
A computation by Hitchin in \cite{Hit1} determines the splitting type of the normal bundle and can be interpreted as follows. Elements $\tau$ of $GL_2({\mathbb C})$ with $\det\tau=-1$ and $\operatorname{tr}\tau =0$ satisfy $\tau^2=1$. For any pair $p_1,p_2$ of quadratic polynomials there exists such an $\tau$ exchanging $p_1$ and $p_2$, which, consequently, acts on $Z$.
The normal bundle of a $\tau$-invariant section splits as ${\mathcal{O}}(2)\oplus {\mathcal{O}}$; otherwise as ${\mathcal{O}}(1)\oplus{\mathcal{O}}(1)$.
\par
To see this directly, observe that modulo translations and the action of $GL_2({\mathbb C})$, $p_1(\zeta)=\zeta$ and $p_2(\zeta)=-\zeta$. The involution $\tau$ is then simply $\zeta\mapsto -\zeta$, and since the normal bundle $N$ of an invariant section satisfies $\tau^\ast N=N$, it must split into line bundles of even degrees. The $\tau$-invariant sections of $Z$ are given by
$z=a\zeta^2 +c$ and by
$$ x=A\left(\zeta+\frac{1+\sqrt{ac}}{a}\right)\left(\zeta-\frac{1-\sqrt{ac}}{a}\right), \enskip y=B\left(\zeta-\frac{1+\sqrt{ac}}{a}\right)\left(\zeta +\frac{1-\sqrt{ac}}{a}\right),$$
where $AB=a^2$. Consequently, for every $\zeta\neq 0,\infty$, the map given by intersecting a section with the fibre $\pi^{-1}(\zeta)$ remains surjective when restricted to sections with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$.
\label{CEH}\end{example}
\begin{example} (${\mathbb{R}} \rm P^4$) Let $Z={\mathbb{P}}\bigl({\mathcal{O}}(1)\oplus{\mathcal{O}}(1)\oplus{\mathcal{O}}\bigr)$ be the compactification of the twistor space of ${\mathbb{R}}^4$. Sections are described as projective equivalence classes:
$$\zeta\mapsto [a_1\zeta+b_1,a_2\zeta+b_2,c],\enskip a_i,b_i,c\in {\mathbb C},\enskip c\neq 0\operatorname{Im}plies \det\begin{pmatrix} a_1 & b_1\\a_2 & b_2\end{pmatrix}\neq 0.$$
The normal bundle of such a section is ${\mathcal{O}}(1)\oplus{\mathcal{O}}(1)$ if $c\neq 0$, and ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ if $c=0$. The manifold of all sections is an open subset of ${\mathbb C}\rm P^4$, and the manifold of real sections, i.e. satisfying $b_1=-\bar a_2$, $b_2=\bar a_1$, $c=\bar c$, is ${\mathbb{R}} \rm P^4$.
\par
Observe that the twistor lines with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ are all contained in their own minitwistor space ${\mathbb{P}}\bigl({\mathcal{O}}(1)\oplus{\mathcal{O}}(1)\oplus 0\bigr)\simeq {\mathbb{P}}^1\times {\mathbb{P}}^1$.
\label{proj}\end{example}
From the point of view of hyperk\"ahler geometry, the difference between the two examples is clear: in the first case, there is a well defined ${\mathcal{O}}(2)$-valued symplectic form along the fibres of $Z$. In the second example, this is not the case. The second example does not fit into Hitchin's theory of folded hyperk\"ahler manifolds: the $3$-dimensional submanifold of real twistor lines with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ is ${\mathbb{R}} \rm P^3$, so it is not even a contact manifold.
\par
Our aim now is to investigate both the common features and the differences in the behaviour of the hypercomplex structure and of the Levi-Civita (i.e. Obata) connection.
{\mathfrak s \mathfrak u}bsection{$2$-Kronecker structures\label{Kr}}
Let $Z$ be a complex manifold of dimension $n+1$ and $\pi:Z\to {\mathbb{P}}^1$ a surjective holomorphic submersion. We are interested in the (necessarily smooth) parameter space $M$ of sections of $\pi$ with normal bundle $N$ isomorphic to $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$, where $k_i\in\{0,1,2\}$ and $n={\mathfrak s \mathfrak u}m k_i$. Its dimension (as long as it is nonempty) is $2n$ and we consider a connected component $M$ which contains a section with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$.
\par
The tangent space $T_m M$ at any $m\in M$ is canonically isomorphic to $H^0(s_m,N)$, where $s_m$ is the section corresponding to $m$. Similarly, we have a rank $n$ bundle $E$ over $M$, the fibre of which is $H^0(s_m,N(-1))$, where $N(-1)=N\otimes \pi^\ast{\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. The multiplication map $H^0(N(-1))\otimes H^0({\mathcal{O}}(1))\to H^0(N)$ induces a homomorphism
$$ \alpha:E\otimes {\mathbb C}^2\to TM,$$
which is an isomorphism at any $m$ with $N_{s_m/Z}\simeq {\mathcal{O}}(1)^n$. It follows that the subset of $M$ consisting of sections with other normal bundles is a divisor ${\partial}elta$ in $M$. We shall assume throughout that the set of singular points of ${\partial}elta$ has codimension $2$ in $M$ (in particular ${\partial}elta$ is reduced). This means that the normal bundle of a section corresponding to a smooth point of ${\partial}elta$ is isomorphic to ${\mathcal{O}}(1)^{n-2}\oplus {\mathcal{O}}(2)\oplus{\mathcal{O}}$.
\par
Observe also that $\alpha$ is injective on each subbundle of the form $E\otimes v$, where $v$ is a fixed nonzero vector in ${\mathbb C}^2$. The image $D_v$ of the subbundle $E\otimes v$ is an integrable distribution on $TM$ (sections of $\pi$ vanishing at the zero of $v\in H^0({\mathcal{O}}(1))$) and we recover $Z$ as the space of leaves of the distribution $D$ on $M\times {\mathbb{P}}^1$ given by $D|_{M\times [v]}=D_v$.
\begin{remark} We can also define $E$ as the kernel of the evaluation map $H^0(N)\otimes{\mathcal{O}}_{{\mathbb{P}}^1}\to N$ (which is what we do in \cite{BP1}), i.e.
$$0\to E_m\otimes {\mathcal{O}}_{{\mathbb{P}}^1}(-1) \stackrel{A}{\longrightarrow} H^0(N)\otimes{\mathcal{O}}_{{\mathbb{P}}^1}\longrightarrow N\to 0.$$
We obtain again a map $\alpha:E\otimes {\mathbb C}^2\to TM$ by restricting $A$ to each subspace of the form $E\otimes\langle v\rangle$, $v\in {\mathbb C}^2$. But then the above sequence identifies $H^0(N(-1))$ with $E_m\otimes H^1({\mathcal{O}}_{{\mathbb{P}}^1}(-2)$. Thus, viewing $\alpha$ as the multiplication map $H^0(N(-1))\otimes H^0({\mathcal{O}}(1))\to H^0(N)$ means that we have implicitly identified $H^1({\mathcal{O}}_{{\mathbb{P}}^1}(-2))$ with
${\mathbb C}$. Such an identification yields also a choice of a symplectic form on $H^0({\mathcal{O}}(1))$ within its conformal class, i.e. an identification of ${\mathbb C}^2$ with $({\mathbb C}^2)^\ast$.
\label{subtle}\end{remark}
This geometric structure on $M$ was introduced in \cite{BP1} as an {\em integrable $2$-Kronecker structure}. We now want to present a different point of view, directly in terms of the tangent bundle of $M$.
\par
Let $M$ be a complex manifold and let ${\partial}elta$ be a divisor satisfying the above smoothness assumption. Suppose that we are given a codimension $1$ distribution ${\mathcal{V}}$ on the smooth locus ${\partial}elta_{\rm reg}$ of ${\partial}elta$. We define $TM(-{\mathcal{V}})$ to be the sheaf of germs of holomorphic vector fields $X$ on $M$ such that $X_x\in {\mathcal{V}}_x$ for any $x\in {\partial}elta_{\rm reg}$. If the sheaf $TM(-{\mathcal{V}})$ is locally free, i.e. a vector bundle $F$, then we obtain a homomorphism $\alpha:F\to TM$ from the inclusion $TM(-{\mathcal{V}}){\mathfrak h}ookrightarrow TM$ (and ${\mathcal{V}}=\operatorname{Im}\alpha$). In the case of $M$ arising as the parameter space of sections as at the beginning of the subsection, $F\simeq E\otimes {\mathbb C}^2$ and the action of $\operatorname{Mat}_{2}({\mathbb C})$ gives an action of complexified quaternions on $TM(-{\mathcal{V}})$. Moreover, for any ${\mathcal{J}}\in SL_2({\mathbb C})$ with $\operatorname{tr} {\mathcal{J}}=0$ (which implies ${\mathcal{J}}^2=-1$), the $i$-eigensubsheaf of $TM(-{\mathcal{V}})$ is closed under the Lie bracket.
Restricting to a real submanifold of $M$ (and corresponding real slices of ${\partial}elta$ and ${\mathcal{V}}$) describes the extension of the hypercomplex or split-hypercomplex geometry to manifolds of sections with jumping normal bundles.
{\mathfrak s \mathfrak u}bsection{Logarithmic hypercomplex structures}
In the setting of the above paragraph, the case of particular interest is ${\mathcal{V}}=T{\partial}elta_{\rm reg}$. Vector fields in $TM(-{\mathcal{V}})$ are called then {\em logarithmic} and $TM(-{\mathcal{V}})$ is denoted by $TM(-\log {\partial}elta)$ \cite{Saito}. Another way to characterise logarithmic vector fields is via the condition $X.z\in (z)$, where $z=0$ is the local equation of ${\partial}elta$. This shows, in particular, that the subsheaf $TM(-\log {\partial}elta)$ is closed under the Lie bracket.
\begin{definition} Let $M$ be a complex manifold and ${\partial}elta$ a divisor in $M$ such that its set of singular points is of codimension $2$ in $M$. A {\em logarithmic biquaternionic structure} on $M$ is an action of $\operatorname{Mat}_2({\mathbb C})$ on $TM(-\log {\partial}elta)$ such that the Nijenhuis tensor of each $A\in \operatorname{Mat}_2({\mathbb C})$ vanishes.\label{log}\end{definition}
\begin{remark} The same definition can be used for real manifolds and we can speak of logarithmic hypercomplex or logarithmic split hypercomplex structures.\end{remark}
Observe that for a logarithmic biquaternionic structure the leaves of the distribution $D_v=\alpha(E\otimes v)$ on ${\partial}elta$ are contained in ${\partial}elta$, i.e. the image of ${\partial}elta$ in each fibre of the twistor space has codimension $1$. In other words $Z$ is a (partial) compactification of the twistor space of a hypercomplex manifold.
More precisely:
\begin{proposition} The following two conditions are equivalent:
\begin{itemize}
\item[(i)] $\operatorname{Im}\alpha_x=T_x{\partial}elta$ for each $x\in {\partial}elta_{\rm reg}$;
\item[(ii)] for each $\zeta\in{\mathbb{P}}^1$, the map ${\partial}elta\to \pi^{-1}(\zeta)$, given by intersecting a section with the fibre, maps a neighbourhood of each point $x\in {\partial}elta_{\rm reg}$ onto an $(n-1)$-dimensional submanifold.
\end{itemize}
\label{Tw}\end{proposition}
\begin{proof} Let $f$ denote the map $M\to \pi^{-1}(\zeta)$, given by intersecting a section with the fibre. For an $x\in{\partial}elta_{\rm reg}$, we have
\begin{equation} \operatorname{Im}\alpha_x=H^0({\mathcal{O}}(1)^{n-2}\oplus {\mathcal{O}}(2)){\mathfrak s \mathfrak u}bset H^0(N)\simeq T_xM.\label{sD}\end{equation}
Thus $df(\operatorname{Im}\alpha_x)$ is an $n-1$-dimensional subspace for any $x\in {\partial}elta_{\rm reg}$.
Since $f$ is a submersion at $x$, the condition $\operatorname{Im}\alpha_x=T_x{\partial}elta$ implies now that the $f({\partial}elta_{\rm reg})$ is an immersed $(n-1)$-dimensional submanifold. Conversely, suppose that the condition (ii) holds. Then the image $f(U)$ of a neighbourhood $U$ of $x\in {\partial}elta_{\rm reg}$ is a codimension $1$ submanifold $Z_0$ of $Z$. It follows that $T_x U\simeq H^0(N_{s/Z_0})$, where $s$ is the section corresponding to $x$. Suppose that $N_{s/Z_0}\simeq \bigoplus_{i=1}^{n-1}{\mathcal{O}}(k_i)$. Given the injection $N_{s/Z_0}{\mathfrak h}ookrightarrow N_{s/Z}$, we have (after reordering the $k_i$) $k_1\leq 2$ and $k_2,\dots,k_{n-1}\leq 1$. Since $H^1(N_{s/Z_0})=0$, we have $\dim {\partial}elta_{\rm reg}=h^0(N_{s/Z_0})$ and therefore ${\mathfrak s \mathfrak u}m_{i=1}^{n-1}(k_i+1)=2n-1$. Thus $(2-k_1)+{\mathfrak s \mathfrak u}m_{i=2}^{n-1}(1-k_i)=0$ and since each summand is nonnegative, we conclude that $N_{s/Z_0}\simeq {\mathcal{O}}(2)\oplus {\mathcal{O}}(1)^{n-2}$.
Thus $T_{x}{\partial}elta=\operatorname{Im}\alpha_{x}$.
\end{proof}
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\begin{remark} This is precisely the situation in Example \operatorname{Re}f{proj}. In Example \operatorname{Re}f{CEH}, the $2$-Kronecker structure is not logarithmic.
\end{remark}
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\section{The meromorphic Obata connection}
The Obata connection of a hypercomplex manifold is the unique torsion-free connection with respect to which the hypercomplex structure is parallel. In the case of a hyperk\"ahler manifold, it coincides with the Levi-Civita connection. From the twistor point of view it is obtained via the Ward transform \cite{Ward,HM}. We now wish to discuss an extension of the Obata connection to a $2$-Kronecker manifold.
\par
Let $Z,M,{\partial}elta,E$ and $\alpha$ be all as in the previous section. We consider the double fibration
$$ M\stackrel{\tau}{\longleftarrow} M\times {\mathbb{P}}^1 \stackrel{\eta}{\longrightarrow} Z.$$
The normal bundle $N$ of any section of $\pi$ is isomorphic to the vertical tangent bundle $T_\pi Z=\operatorname{Ker} d\pi$ restricted to the section, and, consequently, the (holomorphic) tangent bundle $TM$ can be viewed as the Ward transform of $T_\pi Z$, i.e. $TM\simeq \tau_\ast\eta^\ast T_\pi Z$. Similarly, the bundle $E$ is the Ward transform of $T_\pi Z\otimes \pi^\ast{\mathcal{O}}(-1)$.
\par
In \cite[\S 2]{BP1} we have identified the algebraic condition satisfied by the differential operator produced by the Ward transform from any $M$-uniform vector bundle $F$ on $Z$.
In our situation, we can state the results for $F=T_\pi Z(-1)$ as:
\begin{proposition} The bundle $E$ is equipped with a first order differential operator $D:E\to E^\ast\otimes TM$ which satisfies
$D(fs)=\sigma(df\otimes s)+fDs$, where $\sigma$ (the principal symbol of $D$) is the composition of the following two maps
\begin{equation}\begin{CD}
T^\ast M\otimes E @> \alpha^\ast\otimes 1 >> E^\ast\otimes {\mathbb C}^2\otimes E @> 1\otimes\alpha>> E^\ast\otimes TM
\end{CD}\label{phi}\end{equation}
(where ${\mathbb C}^2\simeq ({\mathbb C}^2)^\ast$ as explained in Remark \operatorname{Re}f{subtle}).{\mathfrak h}fill ${\mathbb{B}}ox$
\label{D}\end{proposition}
\begin{remark} On $M\backslash{\partial}elta$\; $\sigma$ is invertible and $\sigma^{-1}\circ D$ is the standard hyperholomorphic connection on $E$, i.e. its tensor product with the standard flat connection on ${\mathbb C}^2$ is the (holomorphic) Obata connection on $M\backslash{\partial}elta$.
\end{remark}
\begin{remark} Given any first order differential operator $D:E\to F$ between (sections of) vector bundles on a manifold $M$, with symbol $\sigma:E\otimes T^\ast M \to F$, we can ``tensor" it with any connection $\nabla$ on a vector bundle $W$ over $M$:
$$ (D\otimes_\sigma \nabla) (e\otimes w)=D(e)\otimes w + (\sigma\otimes 1)(e\otimes \nabla w).$$
The symbol of this new operator is $\sigma\otimes 1$. We can do this for our operator $D$ and the flat connection on ${\mathbb C}^2$. We obtain a differential operator $\tilde D:E\otimes {\mathbb C}^2\to E^\ast\otimes TM\otimes {\mathbb C}^2$ which extends the Obata connection. \label{tensor}\end{remark}
\begin{remark} The results claimed by Pantillie \cite{Pant} would imply that the Obata connection extends to a differential operator satisfying $\tilde D(fs)=\alpha^\ast(df)\otimes s+f\tilde Ds$, but we have trouble following his arguments (in particular the second last paragraph in the proof of his Theorem 2.1).
\end{remark}
We can view $\sigma^{-1}\circ D$ as a meromorphic connection on $E$, with polar set ${\partial}elta$. Similarly the Obata connection on $M\backslash {\partial}elta$ can be viewed as a meromorphic connection on $TM$ with polar set ${\partial}elta$. It follows from Proposition \operatorname{Re}f{D} that $\sigma^{-1}$ generally has a double pole along ${\partial}elta$ and, hence, so does $\sigma^{-1}\circ D$. We shall now discuss conditions under which the pole becomes simple.
\par
Let $z=0$ be the local equation of ${\partial}elta$. The meromorphic connection $\sigma^{-1}\circ D$ has a simple pole if $\lim_{z\to 0} z^2\sigma^{-1}\circ D=0$. Let us trivialise locally $E$, so that $\alpha$ is an endomorphism of the trivial bundle. We can then write $z=\det\alpha$, and owing to Proposition \operatorname{Re}f{D}, we have:
$$ z^2\sigma^{-1}( De)=((\alpha^\ast)_{\rm adj}\otimes 1)(1\otimes \alpha_{\rm adj})(De),$$
where the subscript ``adj" denotes the classical adjoint. Thus, we can conclude:
\begin{lemma} The meromorphic connection $\sigma^{-1}\circ D$ has a simple pole along ${\partial}elta$ provided that $De|_x\in E^\ast_x\otimes\operatorname{Im}\alpha_x$ for any $x\in {\partial}elta_{\rm reg}$ and any local section $e$ of $E$. If this is the case, then the residue of $\sigma^{-1}\circ D$ belongs to $\operatorname{Ker}\alpha^\ast\otimes \operatorname{End} E$.{\mathfrak h}fill${\mathbb{B}}ox$\label{lemma}\end{lemma}
Returning to the description of a $2$-Kronecker structure given at the end of \S\operatorname{Re}f{Kr}, observe that the subsheaf $TM[-{\mathcal{V}}]$ is precisely the subsheaf $\operatorname{Im}\alpha$, and therefore the condition of the last lemma is equivalent to the existence of a differential operator
$$ D^\prime:E\to E^\ast \otimes {\mathbb C}^2\otimes E$$
such that $D=(1\otimes\alpha)\circ D^\prime$.
We shall now show that for a logarithmic hypercomplex structure (Definition \operatorname{Re}f{log}) the condition of the above lemma is automatically satisfied.
\begin{proposition} Suppose that $\operatorname{Im}\alpha_x=T_x{\partial}elta$ for each $x\in {\partial}elta_{\rm reg}$. Then the condition of Lemma \operatorname{Re}f{lemma} is satisfied.\end{proposition}
\begin{proof} Proposition \operatorname{Re}f{Tw} implies that points of ${\partial}elta_{\rm reg}$ correspond to sections of $\pi:Z\to {\mathbb{P}}^1$ contained in a codimension $1$ submanifold $Z^\prime$ of $Z$.
The differential operator $D$ is obtained by the push-forward of the flat relative connection $\nabla_\eta$ on $\eta^\ast T_\pi Z(-1)$, i.e. of the exterior derivative in the vertical directions of the projection $\eta:M\times {\mathbb{P}}^1\to Z$. It follows that, over ${\partial}elta_{\rm reg}$, $D$ restricts to an operator $D^\prime$ defined in the same way as $D$, but with $Z$ replaced by $Z^\prime$. This means that $D^\prime$ takes values in $E\otimes T{\partial}elta_{\rm reg}$.
\end{proof}
Recall that a meromorphic connection on a vector bundle $E$ is called logarithmic, if it has a simple pole along ${\partial}elta=\{z=0\}$ and its residue is of the form $Adz$, where
$A\in\operatorname{End} E$. Thus, under the assumption of the last proposition, $\sigma^{-1}\circ D$ is a logarithmic connection.
\par
We finish with some remarks about the meromorphic Obata connection. As remarked in \operatorname{Re}f{tensor}, we can tensor $D$ with the flat connection on ${\mathbb C}^2$ to obtain an operator $\tilde D:E\otimes {\mathbb C}^2\to E^\ast\otimes TM\otimes {\mathbb C}^2$. The meromorphic Obata connection is a meromorphic connection on $TM$ given
by $(\sigma\otimes 1)^{-1}\circ \tilde D\circ \alpha^{-1}$, where $\sigma$ is the symbol of $D$. Thus, in general we can expect the Obata connection to have a third order pole along the divisor ${\partial}elta$ ($\sigma^{-1}$ contributing two orders and $\alpha^{-1}$ another one). The next example shows that this is indeed the case.
\begin{example} Consider again the twistor space of the Calabi-Eguchi-Hanson gravitational instanton, described in Example \operatorname{Re}f{CEH}. Choose a family of real sections containing the (real) jumping lines. The resulting metric is given in the complex coordinates corresponding to the complex structure $I$, namely $z=-\bar{a}$, $u=\ln \bar A^2$ by the formula \cite[(4.6)]{Hit1}:
$$ {\mathfrak g}amma dzd\bar z+(du+\bar\delta dz)(d\bar u+\delta d\bar z),$$
where ${\mathfrak g}amma$ and $\delta$ are certain functions of the coordinates and the fold ${\partial}elta$ is given by ${\mathfrak g}amma=0$. The Hermitian matrix of this metric is then
$$\begin{pmatrix} {\mathfrak g}amma+{\mathfrak g}amma^{-1}|\delta|^2 & {\mathfrak g}amma^{-1}\bar\delta\\ {\mathfrak g}amma^{-1}\delta & {\mathfrak g}amma^{-1}\end{pmatrix},$$
and it follows that the Levi-Civita connection has poles of third order along ${\partial}elta$.
\end{example}
It is interesting to observe that if the assumption of Lemma \operatorname{Re}f{lemma} is satisfied, then the Obata connection still has a simple pole along ${\partial}elta$ (rather than a second order one, as one could expect). Indeed, as noted above, the operator $D$ is then of the form $D=(1\otimes\alpha)\circ D^\prime$, where $ D^\prime:E\to E^\ast\otimes {\mathbb C}^2 \otimes E$ has symbol $\alpha^\ast\otimes 1$. It follows that the meromorphic Obata connection as an operator $TM\to T^\ast M\otimes TM$ is of the form
$$ (1\otimes\alpha)\circ(\alpha^\ast\otimes 1)^{-1}\circ \widetilde{D^\prime}\circ \alpha^{-1}.$$
Since $(\alpha^\ast\otimes 1)^{-1}\circ \widetilde{D^\prime}$ is a meromorphic connection on $E\otimes{\mathbb C}^2$ with a simple pole along ${\partial}elta$, the meromorphic Obata connection also has a simple pole, as the conjugation by $\alpha$ does not increase the order of the pole.
In particular:
\begin{corollary} Suppose that the equivalent conditions of Proposition \operatorname{Re}f{Tw} are satisfied. Then the holomorphic Obata connection on $M\backslash{\partial}elta$ extends to a logarithmic connection on $M$.{\mathfrak h}fill ${\mathbb{B}}ox$
\end{corollary}
\end{document}
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\begin{document}
\title{A Weak Galerkin finite element method for second-order elliptic problems}
\begin{abstract}
In this paper, authors shall introduce a finite element method by
using a weakly defined gradient operator over discontinuous
functions with heterogeneous properties. The use of weak gradients
and their approximations results in a new concept called {\em
discrete weak gradients} which is expected to play important roles
in numerical methods for partial differential equations. This
article intends to provide a general framework for operating
differential operators on functions with heterogeneous properties.
As a demonstrative example, the discrete weak gradient operator is
employed as a building block to approximate the solution of a model
second order elliptic problem, in which the classical gradient
operator is replaced by the discrete weak gradient. The resulting
numerical approximation is called a weak Galerkin (WG) finite
element solution. It can be seen that the weak Galerkin method
allows the use of totally discontinuous functions in the finite
element procedure. For the second order elliptic problem, an optimal
order error estimate in both a discrete $H^1$ and $L^2$ norms are
established for the corresponding weak Galerkin finite element
solutions. A superconvergence is also observed for the weak Galerkin
approximation.
\end{abstract}
\begin{keywords}
Galerkin finite element methods, discrete gradient, second-order
elliptic problems, mixed finite element methods
\end{keywords}
\begin{AMS}
Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
\end{AMS}
\pagestyle{myheadings}
\section{Introduction}
The goal of this paper is to introduce a numerical approximation
technique for partial differential equations based on a new
interpretation of differential operators and their approximations.
To illustrate the main idea, we consider the Dirichlet problem for
second-order elliptic equations which seeks an unknown functions
$u=u(x)$ satisfying
\begin{eqnarray}
-\nabla\cdot (a \nabla u)+\nabla\cdot (b u)+cu &=& f\quad
\mbox{in}\;\Omega,\label{pde}\\
u&=&g\quad \mbox{on}\; \partial\Omega,\label{bc}
\end{eqnarray}
where $\Omega$ is a polygonal or polyhedral domain in
$\mathbb{R}^d\; (d=2,3)$, $a=(a_{ij}(x))_{d\times d}\in
[L^{\infty}(\Omega)]^{d^2}$ is a symmetric matrix-valued function,
$b=(b_i(x))_{d\times 1}$ is a vector-valued function, and $c=c(x)$
is a scalar function on $\Omega$. Assume that the matrix $a$
satisfies the following property: there exists a constant $\alpha
>0$ such that
\begin{equation}\label{matrix}
\alpha\xi^T\xi\leq \xi^T a\xi,\quad\forall \xi\in \mathbb{R}^d.
\end{equation}
For simplicity, we shall concentrate on two-dimensional problems
only (i.e., $d=2$). An extension to higher-dimensional problems is
straightforward.
The standard weak form for (\ref{pde}) and (\ref{bc}) seeks $u\in
H^1(\Omega)$ such that $u=g$ on $\partial\Omega$ and
\begin{eqnarray}\label{weakform}
(a\nabla u, \nabla v)-(b u,\nabla v)+(cu,v)=(f,v)\quad \forall v\in
H_0^1(\Omega),
\end{eqnarray}
where $(\phi,\psi)$ represents the $L^2$-inner product of
$\phi=\phi(x)$ and $\psi=\psi(x)$ -- either vector-valued or
scalar-valued functions. Here $\nabla u$ denotes the gradient of the
function $u=u(x)$, and $\nabla$ is known as the gradient operator.
In the standard Galerkin method (e.g., see \cite{ci, sue}), the
trial space $H^1(\Omega)$ and the test space $H_0^1(\Omega)$ in
(\ref{weakform}) are each replaced by properly defined subspaces of
finite dimensions. The resulting solution in the subspace/subset is
called a Galerkin approximation. A key feature in the Galerkin
method is that the approximating functions are chosen in a way that
the gradient operator $\nabla$ can be successfully applied to them
in the classical sense. A typical implication of this property in
Galerkin finite element methods is that the approximating functions
(both trial and test) are continuous piecewise polynomials over a
prescribed finite element partition for the domain, often denoted by
${\cal T}_h$. Therefore, a great attention has been paid to a
satisfaction of the embedded ``continuity" requirement in the
research of Galerkin finite element methods in existing literature
till recent advances in the development of discontinuous Galerkin
methods. But the interpretation of the gradient operator still lies
in the classical sense for both ``continuous" and ``discontinuous"
Galerkin finite element methods in current existing literature.
In this paper, we will introduce a weak gradient operator defined on
a space of functions with heterogeneous properties. The weak
gradient operator will then be employed to discretize the problem
(\ref{weakform}) through the use of a discrete weak gradient
operator as building bricks. The corresponding finite element method
is called {\em weak Galerkin} method. Details can be found in
Section \ref{section4}.
To explain weak gradients, let $K$ be any polygonal domain with
interior $K^0$ and boundary $\partial K$. A {\em weak function} on
the region $K$ refers to a vector-valued function $v=\{v_0, v_b\}$
such that $v_0\in L^2(K)$ and $v_b\in H^{\frac12}(\partial K)$. The
first component $v_0$ can be understood as the value of $v$ in the
interior of $K$, and the second component $v_b$ is the value of $v$
on the boundary of $K$. Note that $v_b$ may not be necessarily
related to the trace of $v_0$ on $\partial K$ should a trace be
defined. Denote by $W(K)$ the space of weak functions associated
with $K$; i.e.,
\begin{equation}\label{hi.888.new}
W(K) = \{v=\{v_0, v_b \}:\ v_0\in L^2(K),\; v_b\in
H^{\frac12}(\partial K)\}.
\end{equation}
Recall that the dual of $L^2(K)$ can be identified with itself by
using the standard $L^2$ inner product as the action of linear
functionals. With a similar interpretation, for any $v\in W(K)$, the
{\bf weak gradient} of $v$ can be defined as a linear functional
$\nabla_d v$ in the dual space of $H({\rm div},K)$ whose action on
each $q\in H({\rm div},K)$ is given by
\begin{equation}\label{weak-gradient-new}
(\nabla_d v, q) := -\int_K v_0 \nabla\cdot q dK+ \int_{\partial K}
v_b q\cdot{\bf n} ds,
\end{equation}
where ${\bf n}$ is the outward normal direction to $\partial K$. Observe
that for any $v\in W(K)$, the right-hand side of
(\ref{weak-gradient-new}) defines a bounded linear functional on the
normed linear space $H({\rm div}, K)$. Thus, the weak gradient
$\nabla_d v$ is well defined. With the weak gradient operator
$\nabla_d$ being employed in (\ref{weakform}), the trial and test
functions can be allowed to take separate values/definitions on the
interior of each element $T$ and its boundary. Consequently, we are
left with a greater option in applying the Galerkin to partial
differential equations.
Many numerical methods have been developed for the model problem
(\ref{pde})-(\ref{bc}). The existing methods can be classified into
two categories: (1) methods based on the primary variable $u$, and
(2) methods based on the variable $u$ and a flux variable (mixed
formulation). The standard Galerkin finite element methods
(\cite{ci, sue, baker}) and various interior penalty type
discontinuous Galerkin methods (\cite{arnold, abcm, bo, rwg, rwg-1})
are typical examples of the first category. The standard mixed
finite elements (\cite{rt, arnold-brezzi, babuska, brezzi, bf, bdm,
bddf, wang}) and various discontinuous Galerkin methods based on
both variables (\cite{ccps, cs, cgl, jp}) are representatives of the
second category. Due to the enormous amount of publications
available in general finite element methods, it is unrealistic to
list all the key contributions from the computational mathematics
research community in this article. The main intention of the above
citation is to draw a connection between existing numerical methods
with the one that is to be presented in the rest of the Sections.
The weak Galerkin finite element method, as detailed in Section
\ref{section4}, is closely related to the mixed finite element
method (see \cite{rt, arnold-brezzi, babuska, brezzi, bdm, wang})
with a hybridized interpretation of Fraeijs de Veubeke \cite{fdv1,
fdv2}. The hybridized formulation introduces a new term, known as
the Lagrange multiplier, on the boundary of each element. The
Lagrange multiplier is known to approximate the original function
$u=u(x)$ on the boundary of each element. The concept of {\em weak
gradients} shall provide a systematic framework for dealing with
discontinuous functions defined on elements and their boundaries in
a near classical sense. As far as we know, the resulting weak
Galerkin methods and their error estimates are new in many
applications.
\section{Preliminaries and Notations}\label{section2}
We use standard definitions for the Sobolev spaces $H^s(D)$ and
their associated inner products $(\cdot,\cdot)_{s,D}$, norms
$\|\cdot\|_{s,D}$, and seminorms $|\cdot|_{s,D}$ for $s\ge 0$. For
example, for any integer $s\ge 0$, the seminorm $|\cdot|_{s, D}$
is given by
$$
|v|_{s, D} = \left( \sum_{|\alpha|=s} \int_D |\partial^\alpha v|^2 dD
\right)^{\frac12},
$$
with the usual notation
$$
\alpha=(\alpha_1, \alpha_2), \quad |\alpha| = \alpha_1+\alpha_2,\quad
\partial^\alpha =\partial_{x_1}^{\alpha_1} \partial_{x_2}^{\alpha_2}.
$$
The Sobolev norm $\|\cdot\|_{m,D}$ is given by
$$
\|v\|_{m, D} = \left(\sum_{j=0}^m |v|^2_{j,D} \right)^{\frac12}.
$$
The space $H^0(D)$ coincides with $L^2(D)$, for which the norm and
the inner product are denoted by $\|\cdot \|_{D}$ and
$(\cdot,\cdot)_{D}$, respectively. When $D=\Omega$, we shall drop
the subscript $D$ in the norm and inner product notation. The space
$H({\rm div};\Omega)$ is defined as the set of vector-valued
functions on $\Omega$ which, together with their divergence, are
square integrable; i.e.,
\[
H({\rm div}; \Omega)=\left\{ {\bf v}: \ {\bf v}\in [L^2(\Omega)]^2,
\nabla\cdot{\bf v} \in L^2(\Omega)\right\}.
\]
The norm in $H({\rm div}; \Omega)$ is defined by
$$
\|{\bf v}\|_{H({\rm div}; \Omega)} = \left( \|{\bf v}\|^2 + \|\nabla
\cdot{\bf v}\|^2\right)^{\frac12}.
$$
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\section{Preliminaries and Notations}\label{section2}
We use standard definitions for the Sobolev spaces $H^s(D)$ and
their associated inner products $(\cdot,\cdot)_{s,D}$, norms
$\|\cdot\|_{s,D}$, and seminorms $|\cdot|_{s,D}$ for $s\ge 0$. For
example, for any integer $s\ge 0$, the seminorm $|\cdot|_{s, D}$
is given by
$$
|v|_{s, D} = \left( \sum_{|\alpha|=s} \int_D |\partial^\alpha v|^2 dD
\right)^{\frac12},
$$
with the usual notation
$$
\alpha=(\alpha_1, \alpha_2), \quad |\alpha| = \alpha_1+\alpha_2,\quad
\partial^\alpha =\partial_{x_1}^{\alpha_1} \partial_{x_2}^{\alpha_2}.
$$
The Sobolev norm $\|\cdot\|_{m,D}$ is given by
$$
\|v\|_{m, D} = \left(\sum_{j=0}^m |v|^2_{j,D} \right)^{\frac12}.
$$
The space $H^0(D)$ coincides with $L^2(D)$, for which the norm and
the inner product are denoted by $\|\cdot \|_{D}$ and
$(\cdot,\cdot)_{D}$, respectively. When $D=\Omega$, we shall drop
the subscript $D$ in the norm and inner product notation. The space
$H({\rm div};\Omega)$ is defined as the set of vector-valued
functions on $\Omega$ which, together with their divergence, are
square integrable; i.e.,
\[
H({\rm div}; \Omega)=\left\{ {\bf v}: \ {\bf v}\in [L^2(\Omega)]^2,
\nabla\cdot{\bf v} \in L^2(\Omega)\right\}.
\]
The norm in $H({\rm div}; \Omega)$ is defined by
$$
\|{\bf v}\|_{H({\rm div}; \Omega)} = \left( \|{\bf v}\|^2 + \|\nabla
\cdot{\bf v}\|^2\right)^{\frac12}.
$$
\section{A Weak Gradient Operator and Its Approximation}\label{section3}
The goal of this section is to introduce a weak gradient operator
defined on a space of functions with heterogeneous properties. The
weak gradient operator will then be employed to discretize partial
differential equations. To this end, let $K$ be any polygonal domain
with interior $K^0$ and boundary $\partial K$. A {\em weak function}
on the region $K$ refers to a vector-valued function $v=\{v_0,
v_b\}$ such that $v_0\in L^2(K)$ and $v_b\in H^{\frac12}(\partial
K)$. The first component $v_0$ can be understood as the value of $v$
in the interior of $K$, and the second component $v_b$ is the value
of $v$ on the boundary of $K$. Note that $v_b$ may not be
necessarily related to the trace of $v_0$ on $\partial K$ should a
trace be well defined. Denote by $W(K)$ the space of weak functions
associated with $K$; i.e.,
\begin{equation}\label{hi.888}
W(K) = \{v=\{v_0, v_b \}:\ v_0\in L^2(K),\; v_b\in
H^{\frac12}(\partial K)\}.
\end{equation}
\begin{defi}
The dual of $L^2(K)$ can be identified with itself by using the
standard $L^2$ inner product as the action of linear functionals.
With a similar interpretation, for any $v\in W(K)$, the {\bf weak
gradient} of $v$ is defined as a linear functional $\nabla_d v$ in
the dual space of $H({\rm div},K)$ whose action on each $q\in H({\rm
div},K)$ is given by
\begin{equation}\label{weak-gradient}
(\nabla_d v, q) := -\int_K v_0 \nabla\cdot q dK+ \int_{\partial K}
v_b q\cdot{\bf n} ds,
\end{equation}
where ${\bf n}$ is the outward normal direction to $\partial K$.
\end{defi}
Note that for any $v\in W(K)$, the right-hand side of
(\ref{weak-gradient}) defines a bounded linear functional on the
normed linear space $H({\rm div}, K)$. Thus, the weak gradient
$\nabla_d v$ is well defined. Moreover, if the components of $v$ are
restrictions of a function $u\in H^1(K)$ on $K^0$ and $\partial K$,
respectively, then we would have
$$
-\int_K v_0 \nabla\cdot q dK+ \int_{\partial K} v_b q\cdot{\bf n} ds =
-\int_K u \nabla\cdot q dK+ \int_{\partial K} u q\cdot{\bf n} ds =
\int_K \nabla u \cdot q dK.
$$
It follows that $\nabla_d v= \nabla u$ is the classical gradient of
$u$.
Next, we introduce a discrete weak gradient operator by defining
$\nabla_d$ in a polynomial subspace of $H({\rm div}, K)$. To this
end, for any non-negative integer $r\ge 0$, denote by $P_{r}(K)$ the
set of polynomials on $K$ with degree no more than $r$. Let
$V(K,r)\subset [P_{r}(K)]^2$ be a subspace of the space of
vector-valued polynomials of degree $r$. A discrete weak gradient
operator, denoted by $\nabla_{d,r}$, is defined so that
$\nabla_{d,r} v \in V(K,r)$ is the unique solution of the following
equation
\begin{equation}\label{discrete-weak-gradient}
\int_K \nabla_{d,r} v\cdot q dK = -\int_K v_0 \nabla\cdot q dK+
\int_{\partial K} v_b q\cdot{\bf n} ds,\qquad \forall q\in V(K,r).
\end{equation}
It is not hard to see that the discrete weak gradient operator
$\nabla_{d,r}$ is a Galerkin-type approximation of the weak gradient
operator $\nabla_d$ by using the polynomial space $V(K,r)$.
The classical gradient operator $\nabla=(\partial_{x_1},
\partial_{x_2})$ should be applied to functions with certain smoothness in
the design of numerical methods for partial differential equations.
For example, in the standard Galerkin finite element method, such a
``smoothness" often refers to continuous piecewise polynomials over
a prescribed finite element partition. With the weak gradient
operator as introduced in this section, derivatives can be taken for
functions without any continuity across the boundary of each
triangle. Thus, the concept of weak gradient allows the use of
functions with heterogeneous properties in approximation.
Analogies of weak gradient can be established for other differential
operators such as divergence and curl operators. Details for weak
divergence and weak curl operators and their applications in
numerical methods will be given in forthcoming papers.
\section{A Weak Galerkin Finite Element Method}\label{section4}
The goal of this section is to demonstrate how discrete weak
gradients be used in the design of numerical schemes that
approximate the solution of partial differential equations. For
simplicity, we take the second order elliptic equation (\ref{pde})
as a model for discussion. With the Dirichlet boundary condition
(\ref{bc}), the standard weak form seeks $u\in H^1(\Omega)$ such
that $u=g$ on $\partial\Omega$ and
\begin{eqnarray}\label{weakform-new}
(a\nabla u, \nabla v)-(b u,\nabla v)+(cu,v)=(f,v)\quad \forall v\in
H_0^1(\Omega).
\end{eqnarray}
Let ${\cal T}_h$ be a triangular partition of the domain $\Omega$
with mesh size $h$. Assume that the partition ${\cal T}_h$ is shape
regular so that the routine inverse inequality in the finite element
analysis holds true (see \cite{ci}). In the general spirit of
Galerkin procedure, we shall design a weak Galerkin method for
(\ref{weakform-new}) by following two basic principles: {\em (1)
replace $H^1(\Omega)$ by a space of discrete weak functions defined
on the finite element partition ${\cal T}_h$ and the boundary of
triangular elements; (2) replace the classical gradient operator by
a discrete weak gradient operator $\nabla_{d,r}$ for weak functions
on each triangle $T$.} Details are to be presented in the rest of
this section.
For each $T\in {\cal T}_h$, Denote by $P_j(T^0)$ the set of
polynomials on $T^0$ with degree no more than $j$, and
$P_\ell(\partial T)$ the set of polynomials on $\partial T$ with
degree no more than $\ell$ (i.e., polynomials of degree $\ell$ on
each line segment of $\partial T$). A {\em discrete weak function}
$v=\{v_0, v_b\}$ on $T$ refers to a weak function $v=\{v_0, v_b\}$
such that $v_0\in P_j(T^0)$ and $v_b\in P_\ell(\partial T)$ with
$j\ge 0$ and $\ell \ge 0$. Denote this space by $W(T, j, \ell)$,
i.e.,
$$
W(T,j,\ell) := \left\{v=\{v_0, v_b\}:\ v_0\in P_j(T^0), v_b\in
P_\ell(\partial T)\right\}.
$$
The corresponding finite element space would be defined by patching
$W(T,j,\ell)$ over all the triangles $T\in {\cal T}_h$. In other
words, the weak finite element space is given by
\begin{equation}\label{weak-fes}
S_h(j,\ell) :=\left\{ v=\{v_0, v_b\}:\ \{v_0, v_b\}|_{T}\in
W(T,j,\ell), \forall T\in {\cal T}_h \right\}.
\end{equation}
Denote by $S_h^0(j,\ell)$ the subspace of $S_h(j,\ell)$ with
vanishing boundary values on $\partial\Omega$; i.e.,
\begin{equation}\label{weak-fes-homo}
S_h^0(j,\ell) :=\left\{ v=\{v_0, v_b\}\in S_h(j,\ell),
{v_b}|_{\partial T\cap \partial\Omega}=0, \ \forall T\in {\cal T}_h
\right\}.
\end{equation}
According to (\ref{discrete-weak-gradient}), for each $v=\{v_0,
v_b\} \in S_h(j,\ell)$, the discrete weak gradient of $v$ on each
element $T$ is given by the following equation:
\begin{equation}\label{discrete-weak-gradient-new}
\int_T \nabla_{d,r} v\cdot q dT = -\int_T v_0 \nabla\cdot q dT+
\int_{\partial T} v_b q\cdot{\bf n} ds,\qquad \forall q\in V(T,r).
\end{equation}
Note that no specific examples of the approximating space $V(T,r)$
have been mentioned, except that $V(T,r)$ is a subspace of the set
of vector-valued polynomials of degree no more than $r$ on $T$.
For any $w,v \in S_h(j,\ell)$, we introduce the following bilinear
form
\begin{equation}\label{linearform-a}
a(w,v)=(a\nabla_{d,r} w,\;\nabla_{d,r} v)-(b u_0,\nabla_{d,r}
v)+(cu_0,v_0),
\end{equation}
where
\begin{eqnarray*}
(a\nabla_{d,r} w,\;\nabla_{d,r} v)&=&\int_\Omega a\nabla_{d,r} w
\cdot\nabla_{d,r}v d\Omega,\\
(bw_0,\;\nabla_{d,r} v)&=&\int_{\Omega} bu_0\cdot \nabla_{d,r}v
d\Omega,\\
(cw_0,v_0)&=&\int_\Omega cw_0 v_0 d\Omega.
\end{eqnarray*}
\begin{algorithm}
A numerical approximation for (\ref{pde}) and (\ref{bc}) can be
obtained by seeking $u_h=\{u_0,u_b\}\in S_h(j,\ell)$ satisfying
$u_b= Q_b g$ on $\partial \Omega$ and the following equation:
\begin{equation}\label{WG-fem}
a(u_h,v)=(f,\;v_0), \quad\forall\ v=\{v_0, v_b\}\in S_h^0(j,\ell),
\end{equation}
where $Q_b g$ is an approximation of the boundary value in the
polynomial space $P_\ell(\partial T\cap \partial\Omega)$. For
simplicity, $Q_b g$ shall be taken as the standard $L^2$ projection
for each boundary segment; other approximations of the boundary
value $u=g$ can also be employed in (\ref{WG-fem}).
\end{algorithm}
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\section{Examples of Weak Galerkin Method with Properties}
Although the weak Galerkin scheme (\ref{WG-fem}) is defined for
arbitrary indices $j, \ell$, and $r$, the method can be shown to
produce good numerical approximations for the solution of the
original partial differential equation only with a certain
combination of their values. For one thing, there are at least two
prominent properties that the discrete gradient operator
$\nabla_{d,r}$ should possess in order for the weak Galerkin method
to work well. These two properties are:
\begin{enumerate}
\item[\bf P1:] For any $v\in S_h(j,\ell)$, if $\nabla_{d,r} v=0$ on $T$, then
one must have $v\end{equation}uiv constant$ on $T$. In other words,
$v_0=v_b=constant$ on $T$;
\item[\bf P2:] Let $u\in H^m(\Omega) (m\ge 1)$ be a smooth function on
$\Omega$, and $Q_h u$ be a certain interpolation/projection of $u$
in the finite element space $S_h(j,\ell)$. Then, the discrete weak
gradient of $Q_h u$ should be a good approximation of $\nabla u$.
\end{enumerate}
The following are two examples of weak finite element spaces that
fit well into the numerical scheme (\ref{WG-fem}).
\begin{example}\label{wg-example1}
In this example, we take $\ell=j+1, r=j+1$, and
$V(T,j+1)=\left[P_{j+1}(T)\right]^2$, where $j\ge 0$ is any
non-negative integer. Denote by $S_h(j,j+1)$ the corresponding
finite element space. More precisely, the finite element space
$S_h(j,j+1)$ consists of functions $v=\{v_0, v_b\}$ where $v_0$ is a
polynomial of degree no more than $j$ in $T^0$, and $v_b$ is a
polynomial of degree no more than $j+1$ on $\partial T$. The space
$V(T,r)$ used to define the discrete weak gradient operator
$\nabla_{d,r}$ in (\ref{discrete-weak-gradient-new}) is given as
vector-valued polynomials of degree no more than $j+1$ on $T$.
\end{example}
\begin{example}\label{wg-example2}
In the second example, we take $\ell=j, r=j+1$, and
$V(T,r=j+1)=\left[P_{j}(T)\right]^2 + \widehat P_j(T) {\bf x}$, where
${\bf x}=(x_1,x_2)^T$ is a column vector and $\widehat P_j(T)$ is the
set of homogeneous polynomials of order $j$ in the variable ${\bf x}$.
Denote by $S_h(j,j)$ the corresponding finite element space. Note
that the space $V(T,r)$ that was used to define a discrete weak
gradient is in fact the usual Raviart-Thomas element \cite{rt} of
order $j$ for the vector component.
\end{example}
Let us demonstrate how the two properties {\bf P1} and {\bf P2} are
satisfied with the two examples given as above. For simplicity, we
shall present results only for {\bf WG Example \ref{wg-example1}}.
The following result addresses a satisfaction of the property {\bf
P1}.
\begin{lemma}\label{lemma-zero}
For any $v=\{v_0, v_b\}\in W(T, j, j+1)$, let $\nabla_{d,j+1} v$ be
the discrete weak gradient of $v$ on $T$ as defined in
(\ref{discrete-weak-gradient-new}) with
$V(T,r)=\left[P_{j+1}(T)\right]^2$. Then, $\nabla_{d,j+1} v =0$
holds true on $T$ if and only if $v=constant$ (i.e.,
$v_0=v_b=constant$).
\end{lemma}
\begin{proof}
It is trivial to see from (\ref{discrete-weak-gradient-new}) that if
$v=constant$ on $T$, then the right-hand side of
(\ref{discrete-weak-gradient-new}) would be zero for any $q\in
V(T,j+1)$. Thus, we must have $\nabla_{d,j+1} v =0$.
Now assume that $\nabla_{d,j+1} v =0$. It follows from
(\ref{discrete-weak-gradient-new}) that
\begin{equation}\label{discrete-weak-gradient-newv}
-\int_T v_0 \nabla\cdot q dT+ \int_{\partial T} v_b q\cdot{\bf n}
ds=0,\qquad \forall q\in V(T,j+1).
\end{equation}
Let $\bar{v}_0$ be the average of $v_0$ over $T$. Using the results
of \cite{bdm}, there exists a vector-valued polynomial $q_1\in
V(T,j+1)=[P_{j+1}(T)]^2$ such that $q_1\cdot{\bf n}=0$ on $\partial T$
and $\nabla\cdot q_1 = v_0 - \bar{v}_0$. With $q=q_1$ in
(\ref{discrete-weak-gradient-newv}), we arrive at $\int_T
(v_0-\bar{v}_0)^2 dT=0$. It follows that $v_0=\bar{v}_0$, and
(\ref{discrete-weak-gradient-newv}) can be rewritten as
\begin{equation}\label{discrete-weak-gradient-newvw}
\int_{\partial T} (v_b-v_0) q\cdot{\bf n} ds=0,\qquad \forall q\in
V(T,j+1).
\end{equation}
Now since $v_b-v_0\in P_{j+1}(\partial T)$, then one may select a
$q\in V(T,j+1)=[P_{j+1}(T)]^2$ such that
$$
\int_{\partial T} \phi q\cdot{\bf n} ds = \int_{\partial T} \phi
(v_b-v_0)ds,\qquad \forall \phi\in P_{j+1}(\partial T),
$$
which, together with (\ref{discrete-weak-gradient-newvw}) and
$\phi=v_b-v_0$ yields
$$
\int_{\partial T} (v_b-v_0)^2ds = 0.
$$
The last equality implies $v_b=v_0=constant$, which completes a
proof of the lemma.
\end{proof}
To verify property {\bf P2}, let $u\in H^1(T)$ be a smooth function
on $T$. Denote by $Q_h u=\{Q_0 u,\;Q_bu\}$ the $L^2$ projection
onto $P_j(T^0)\times P_{j+1}(\partial T)$. In other words, on each
element $T$, the function $Q_0 u$ is defined as the $L^2$ projection
of $u$ in $P_j(T)$ and on $\partial T$, $Q_b u$ is the $L^2$
projection in $P_{j+1}(\partial T)$. Furthermore, let $R_h$ be the
local $L^2$ projection onto $V(T,j+1)$. According to the definition
of $\nabla_{d,j+1}$, the discrete weak gradient function
$\nabla_{d,j+1}(Q_hu)$ is given by the following equation:
\begin{equation}\label{discrete-weak-gradient-hi}
\int_T \nabla_{d,j+1}(Q_h u) \cdot q dT = -\int_T (Q_0 u)
\nabla\cdot q dT+ \int_{\partial T} (Q_b u) q\cdot{\bf n} ds,\quad
\forall q\in V(K,j+1).
\end{equation}
Since $Q_0$ and $Q_b$ are $L^2$-projection operators, then the
right-hand side of (\ref{discrete-weak-gradient-hi}) is given by
\begin{eqnarray*}
-\int_T (Q_0 u) \nabla\cdot q dT+ \int_{\partial T} (Q_b u)
q\cdot{\bf n} ds &=& -\int_T u \nabla\cdot q dT+ \int_{\partial T} u
q\cdot{\bf n} ds \\
&=& \int_T (\nabla u)\cdot q dT = \int_T (R_h \nabla u)\cdot q dT.
\end{eqnarray*}
Thus, we have derived the following useful identity:
\begin{equation}\label{4.88}
\nabla_{d,j+1}(Q_h u) =R_h (\nabla u),\qquad \forall u\in H^1(T).
\end{equation}
The above identity clearly indicates that $\nabla_{d,j+1}(Q_h u)$ is
an excellent approximation of the classical gradient of $u$ for any
$u\in H^1(T)$. Thus, it is reasonable to believe that the weak
Galerkin finite element method shall provide a good numerical scheme
for the underlying partial differential equations.
\section{Mass Conservation of Weak Galerkin}
The second order elliptic equation (\ref{pde}) can be rewritten in a
conservative form as follows:
$$
\nabla \cdot q + cu = f, \quad q=-a\nabla u + bu.
$$
Let $T$ be any control volume. Integrating the first equation over
$T$ yields the following integral form of mass conservation:
\begin{equation}\label{conservation.01}
\int_{\partial T} q\cdot {\bf n} ds + \int_T cu dT = \int_T f dT.
\end{equation}
We claim that the numerical approximation from the weak Galerkin
finite element method for (\ref{pde}) retains the mass conservation
property (\ref{conservation.01}) with a numerical flux $q_h$. To
this end, for any given $T\in {\cal T}_h$, we chose in
(\ref{WG-fem}) a test function $v=\{v_0, v_b=0\}$ so that $v_0=1$ on
$T$ and $v_0=0$ elsewhere. Using the relation (\ref{linearform-a}),
we arrive at
\begin{equation}\label{mass-conserve.08}
\int_T a\nabla_{d,r} u_h\cdot \nabla_{d,r}v dT - \int_T b u_{0}
\cdot\nabla_{d,r}v dT + \int_T c u_{0} dT = \int_T f dT.
\end{equation}
Using the definition (\ref{discrete-weak-gradient-new}) for
$\nabla_{d,r}$, one has
\begin{eqnarray}
\int_T a\nabla_{d,r} u_h\cdot \nabla_{d,r}v dT &=& \int_T
R_h(a\nabla_{d,r} u_h)\cdot \nabla_{d,r}v dT \nonumber\\
&=& - \int_T \nabla\cdot R_h(a\nabla_{d,r} u_h) dT \nonumber\\
&=& - \int_{\partial T} R_h(a\nabla_{d,r}u_h)\cdot{\bf n} ds
\label{conserv.88}
\end{eqnarray}
and
\begin{eqnarray}
\int_T b u_{0} \cdot\nabla_{d,r}v dT &=& \int_T R_h(b u_{0})
\cdot\nabla_{d,r}v dT\nonumber\\
&=& -\int_T \nabla\cdot R_h(b u_{0})dT\nonumber\\
&=& -\int_{\partial T} R_h(b u_{0})\cdot{\bf n} ds\label{conserve.89}
\end{eqnarray}
Now substituting (\ref{conserve.89}) and (\ref{conserv.88}) into
(\ref{mass-conserve.08}) yields
\begin{equation}\label{mass-conserve.09}
\int_{\partial T} R_h\left(-a\nabla_{d,r}u_h + b u_{0}
\right)\cdot{\bf n} ds +\int_T c u_{0} dT = \int_T f dT,
\end{equation}
which indicates that the weak Galerkin method conserves mass with a
numerical flux given by
$$
q_h\cdot{\bf n} =R_h\left(-a\nabla_{d,r}u_h + b u_{0} \right)\cdot{\bf n}.
$$
The numerical flux $q_h\cdot{\bf n}$ can be verified to be continuous
across the edge of each element $T$ through a selection of the test
function $v=\{v_0,v_b\}$ so that $v_0\end{equation}uiv 0$ and $v_b$ arbitrary.
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\section{Existence and Uniqueness for Weak Galerkin Approximations}
Assume that $u_h$ is a weak Galerkin approximation for the problem
(\ref{pde}) and (\ref{bc}) arising from (\ref{WG-fem}) by using the
finite element space $S_h(j,j+1)$ or $S_h(j,j)$. The goal of this
section is to derive a uniqueness and existence result for $u_h$.
For simplicity, details are only presented for the finite element
space $S_h(j,j+1)$; the result can be extended to $S_h(j,j)$ without
any difficulty.
First of all, let us derive the following analogy of G\aa{rding's}
inequality.
\begin{lemma}
Let $S_h(j,\ell)$ be the weak finite element space defined in
(\ref{weak-fes}) and $a(\cdot,\cdot)$ be the bilinear form given in
(\ref{linearform-a}). There exists a constant $K$ and $\alpha_1$
satisfying
\begin{equation}\label{garding}
a(v,v)+K(v_0, v_0)\ge \alpha_1(\|\nabla_{d,r}v\|^2+\|v_0\|^2),
\end{equation}
for all $v\in S_{h}(j,\ell)$.
\end{lemma}
\begin{proof}
Let $B_1=\|b\|_{L^\infty(\Omega)}$ and
$B_2=\|c\|_{L^\infty(\Omega)}$ be the $L^\infty$ norm of the
coefficients $b$ and $c$, respectively. Since
\begin{eqnarray*}
|(bv_0, \nabla_{d,r} v)|&\le& B_1\|\nabla_{d,r} v\|\ \|v_0\|,\\
|(cv_0, v_0)|&\leq & B_2 \|v_0\|^2,
\end{eqnarray*}
then it follows from (\ref{linearform-a}) that there exists a
constant $K$ and $\alpha_1$ such that
\begin{eqnarray*}
a(v,v)+K(v_0,v_0)&\ge& \alpha\|\nabla_{d,r}v\|^2-B_1\|\nabla_{d,r}v\|\|v_0\|+(K-B_2)\|v_0\|^2\\
&\ge& \alpha_1(\|\nabla_{d,r}v\|^2+\|v_0\|^2),
\end{eqnarray*}
which completes the proof.
\end{proof}
For simplicity of notation, we shall drop the subscript $r$ in the
discrete weak gradient operator $\nabla_{d,r}$ from now on. Readers
should bear in mind that $\nabla_d$ refers to a discrete weak
gradient operator defined by using the setups of either Example
\ref{wg-example1} or Example \ref{wg-example2}. In fact, for these
two examples, one may also define a projection $\Pi_h$ such that
$\Pi_h{\bf q}\in H({\rm div},\Omega)$, and on each $T\in {\cal T}_h$,
one has $\Pi_h{\bf q} \in V(T, r=j+1)$ and the following identity
$$
(\nabla\cdot{\bf q},\;v_0)_T=(\nabla\cdot\Pi_h{\bf q},\;v_0)_T, \qquad
\forall v_0\in P_j(T^0).
$$
The following result is based on the above property of $\Pi_h$.
\begin{lemma}
For any ${\bf q}\in H({\rm div},\Omega)$, we have
\begin{equation}\label{4.200}
\sum_{T\in {\cal T}_h}(-\nabla\cdot{\bf q}, \;v_0)_T=\sum_{T\in {\cal
T}_h}(\Pi_h{\bf q}, \;\nabla_dv)_T,
\end{equation}
for all $v=\{v_0,v_b\}\in S^0_h(j,j+1)$.
\end{lemma}
\begin{proof}
The definition of $\Pi_h$ and the definition of $\nabla_d v$ imply ,
\begin{eqnarray*}
\sum_{T\in {\cal T}_h}(-\nabla\cdot{\bf q}, \;v_0)_T&=&\sum_{T\in {\cal T}_h}(-\nabla\cdot \Pi_h{\bf q}, \;v_0)_T\\
&=&\sum_{T\in {\cal T}_h}(\Pi_h{\bf q}, \nabla_d v)_T - \sum_{T\in {\cal
T}_h} \langle v_b,
\Pi_h {\bf q}\cdot {\bf n}\rangle_{\partial T} \\
&=&\sum_{T\in {\cal T}_h}(\Pi_h {\bf q}, \nabla_d v)_T.
\end{eqnarray*}
Here we have used the fact that $\Pi_h {\bf q}\cdot {\bf n}$ is continuous
across each interior edge and $v_b=0$ on $\partial \Omega$. This
completes the proof.
\end{proof}
\begin{lemma}\label{approx}
For $u\in H^{1+s}(\Omega)$ with $s>0$, we have
\begin{eqnarray}
\|\Pi_h(a\nabla u)-a\nabla_d(Q_h u)\|&\le& Ch^s\|u\|_{1+s},\label{a2}\\
\|\nabla u-\nabla_d(Q_hu)\|&\le&Ch^s\|u\|_{1+s}.\label{a3}
\end{eqnarray}
\end{lemma}
\begin{proof} Since from (\ref{4.88}) we have $\nabla_d(Q_h u) =
R_h(\nabla u)$, then
$$
\|\Pi_h(a\nabla u)-a\nabla_d(Q_h u)\| = \|\Pi_h(a\nabla
u)-aR_h(\nabla u)\|.
$$
Using the triangle inequality and the definition of $\Pi_h$ and
$R_h$, we have
\begin{eqnarray*}
\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|&\le&\|\Pi_h(a\nabla u)-a\nabla u\|+\|a\nabla u-aR_h(\nabla u)\|\\
&\le& Ch^s\|u\|_{1+s}.
\end{eqnarray*}
The estimate (\ref{a3}) can be derived in a similar way. This
completes a proof of the lemma.
\end{proof}
We are now in a position to establish a solution uniqueness and
existence for the weak Galerkin method (\ref{WG-fem}). It suffices
to prove that the solution is unique. To this end, let $e\in
S_h^0(j,j+1)$ be a discrete weak function satisfying
\begin{equation}\label{uniq}
a(e,v)=0,\qquad\forall v=\{v_0, v_b\}\in S_h^0(j,j+1).
\end{equation}
The goal is to show that $e\end{equation}uiv 0$ by using a duality approach
similar to what Schatz \cite{schatz} did for the standard Galerkin
finite element methods.
\begin{lemma}\label{L2byH1}
Let $e=\{e_0, e_b\}\in S_h^0(j,j+1)$ be a discrete weak function
satisfying (\ref{uniq}). Assume that the dual of (\ref{pde}) with
homogeneous Dirichlet boundary condition has the $H^{1+s}$
regularity ($s\in (0,1]$). Then, there exists a constant $C$ such
that
\begin{equation}\label{dual-1}
\|e_0\|\le Ch^s\|\nabla_d e\|,
\end{equation}
provided that the mesh size $h$ is sufficient small, but a fixed
constant.
\end{lemma}
\begin{proof}
Consider the following dual problem: Find $w\in H^1(\Omega)$ such
that
\begin{eqnarray}
-\nabla\cdot (a \nabla w)-b\cdot\nabla w+ cw &=&e_0 \quad
\mbox{in}\;\Omega\label{dual1}\\
w&=&0\quad \mbox{on}\; \partial\Omega,\label{dual1-BC}
\end{eqnarray}
The assumption of $H^{1+s}$ regularity implies that $w\in
H^{1+s}(\Omega)$ and there is a constant $C$ such that
\begin{equation}\label{reg1}
\|w\|_{1+s}\le C\|e_0\|.
\end{equation}
Testing (\ref{dual1}) against $e_0$ and then using (\ref{4.200})
lead to
\begin{eqnarray*}
\|e_0\|^2&=&(-\nabla\cdot (a \nabla w),\;e_0)-(b\cdot\nabla w,\;e_0)+ (cw,\;e_0)\\
&=&(\Pi_h(a\nabla w),\;\nabla_d e)-(\nabla w,\;be_0)+ (cw,\;e_0)\\
&=&(\Pi_h(a\nabla w)-a\nabla_d(Q_h w),\;\nabla_d e)+(a\nabla_d(Q_h w),\;\nabla_d e)\\
& &-(\nabla w-\nabla_d(Q_hw),\;be_0))-(\nabla_d(Q_hw),\;be_0)\\
& &+(cw-c(Q_0w),\;e_0)+(Q_0w,\;ce_0).
\end{eqnarray*}
The sum of the second, forth and sixth term on the right hand side
of the above equation equals $a(e, Q_hw)=0$ due to (\ref{uniq}).
Therefore, it follows from Lemma \ref{approx} that
\begin{eqnarray*}
\|e_0\|^2&=&(\Pi_h(a\nabla w)-a\nabla_d(Q_h w),\;\nabla_d e)-(\nabla w-\nabla_d(Q_hw),\;be_0)\\
& &+(c(w-Q_0w),\;e_0)\\
&\le&Ch^s\|w\|_{1+s}\left(\|\nabla_de\|+\|e_0\|\right) + C h \|w\|_1
\; \|e_0\|.
\end{eqnarray*}
Using the $H^{1+s}$-regularity assumption (\ref{reg1}), we arrive at
$$
\|e_0\|^2 \leq C h^s\|e_0\|\left(\|\nabla_de\|+\|e_0\|\right),
$$
which leads to
$$
\|e_0\| \leq C h^s\left(\|\nabla_de\|+\|e_0\|\right).
$$
Thus, when $h$ is sufficiently small, one would obtain the desired
estimate (\ref{dual-1}). This completes the proof.
\end{proof}
\begin{theorem}
Assume that the dual of (\ref{pde}) with homogeneous Dirichlet
boundary condition has $H^{1+s}$-regularity for some $s\in (0,1]$.
The weak Gakerkin finite element method defined in (\ref{WG-fem})
has a unique solution in the finite element spaces $S_h(j,j+1)$ and
$S_j(j,j)$ if the meshsize $h$ is sufficiently small, but a fixed
constant.
\end{theorem}
\begin{proof} Observe that uniqueness is equivalent to existence for the solution of
(\ref{WG-fem}) since the number of unknowns is the same as the
number of equations. To prove a uniqueness, let $u^{(1)}_h$ and
$u^{(2)}_h$ be two solutions of (\ref{WG-fem}). By letting
$e=u^{(1)}_h- u^{(2)}_h$ we see that (\ref{uniq}) is satisfied. Now
we have from the G{\aa}rding's inequality (\ref{garding}) that
$$
a(e,e) + K \|e_0\| \ge \alpha_1\left( \|\nabla_d e\| +
\|e_0\|\right).
$$
Thus, it follows from the estimate (\ref{dual-1}) of Lemma
\ref{L2byH1} that
$$
\alpha_1\left( \|\nabla_d e\| + \|e_0\|\right)\leq CK h^s \|\nabla_d
e\|
$$
for $h$ being sufficiently small. Now chose $h$ small enough so that
$CKh^s\leq \frac{\alpha_1}{2}$. Thus,
$$
\|\nabla_d e\| + \|e_0\|\ = 0,
$$
which, together with Lemma \ref{lemma-zero}, implies that $e$ is a
constant and $e_0=0$. This shows that $e=0$ and consequently,
$u^{(1)}_h=u^{(2)}_h$.
\end{proof}
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\section{Error Analysis}
The goal of this section is to derive some error estimate for the
weak Galerkin finite element method (\ref{WG-fem}). We shall follow
the usual approach in the error analysis: (1) investigating the
difference between the weak finite element approximation $u_h$ with
a certain interpolation/projection of the exact solution through an
error equation, (2) using a duality argument to analyze the error in
the $L^2$ norm.
Let us begin with the derivation of an error equation for the weak
Galerkin approximation $u_h$ and the $L^2$ projection of the exact
solution $u$ in the weak finite element space $S_h(j,j+1)$. Recall
that the $L^2$ projection is denoted by $Q_h u \end{equation}uiv \{Q_0u, Q_b
u\}$, where $Q_0$ denotes the local $L^2$ projection onto $P_j(T)$
and $Q_b$ is the local $L^2$ projection onto $P_{j+1}(\partial T)$
on each triangular element $T\in {\cal T}_h$. Let $v=\{v_0,v_b\}\in
S_h^0(j,j+1)$ be any test function. By testing (\ref{pde}) against
the first component $v_0$ and using (\ref{4.200}) we arrive at
\begin{eqnarray*}
(f, v_0) &=& \sum_{T\in {\cal T}_h}(-\nabla\cdot (a\nabla u), \;v_0)_T
+(\nabla\cdot (b u),\; v_0)+(cu, \;v_0)\nonumber\\
&=&(\Pi_h(a\nabla u),\; \nabla_dv)-(\Pi_h(bu),\; \nabla_d v)+(cu,\; v_0).
\end{eqnarray*}
Adding and subtracting the term $a(Q_hu, v)\end{equation}uiv
(a\nabla_d(Q_hu),\;\nabla_d
v)-(b(Q_0u),\;\nabla_dv)+(c(Q_0u),\;v_0)$ on the right hand side of
the above equation and then using (\ref{4.88}) we obtain
\begin{eqnarray}
(f, v_0)&=&(a\nabla_d(Q_hu),\;\nabla_d v)-(bQ_0u,\;\nabla_dv)+(cQ_0u,\;v_0)\label{true}\\
& &+(\Pi_h(a\nabla u)-aR_h(\nabla u),\;
\nabla_dv)\nonumber\\
& & -(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0),\nonumber
\end{eqnarray}
which can be rewritten as
\begin{eqnarray}
a(u_h, v)&=&a(Q_hu,v)+(\Pi_h(a\nabla u)-aR_h(\nabla u),\;
\nabla_dv)\nonumber\\
& & -(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0).\nonumber
\end{eqnarray}
It follows that
\begin{eqnarray}
a(u_h-Q_hu,\;v)&=&(\Pi_h(a\nabla u)-aR_h(\nabla u),\; \nabla_dv)\nonumber\\
& &-(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0).\label{diff}
\end{eqnarray}
The equation (\ref{diff}) shall be called the {\em error equation}
for the weak Galerkin finite element method (\ref{WG-fem}).
\subsection{An estimate in a discrete $H^1$-norm}
We begin with the following lemma which provides an estimate for the
difference between the weak Galerkin approximation $u_h$ and the
$L^2$ projection of the exact solution of the original problem.
\begin{lemma}\label{h1-error}
Let $u\in H^{1}(\Omega)$ be the solution of (\ref{pde}) and
(\ref{bc}). Let $u_h\in S_h(j,j+1)$ be the weak Galerkin
approximation of $u$ arising from (\ref{WG-fem}). Denote by
$e_h:=u_h-Q_h u$ the difference between the weak Galerkin
approximation and the $L^2$ projection of the exaction solution
$u=u(x_1,x_2)$. Then there exists a constant $C$ such that
\begin{eqnarray}
\frac{\alpha_1}{2}(\|\nabla_d(e_h)\|^2+\|e_{h,0}\|^2) &\le&
C\left(\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|^2 +\|c(u-Q_0u)\|^2
\right.\nonumber\\
& & +\left.
\|\Pi_h(bu)-bQ_0u\|^2\right)+K\|u_0-Q_0u\|^2.\label{H1errorestimate}
\end{eqnarray}
\end{lemma}
\begin{proof} Substituting $v$ in (\ref{diff}) by $e_h:=u_h-Q_hu$
and using the usual Cauchy-Schwarz inequality we arrive at
\begin{eqnarray*}
a(e_h,\;e_h)&=&(\Pi_h(a\nabla u)-aR_h(\nabla u),\; \nabla_d (u_h-Q_hu))\\
& &-(\Pi_h(bu)-bQ_0u,\; \nabla_d (u_h-Q_hu))+(c(u-Q_0u),\; u_0-Q_0u)\\
&\le&\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|\;\|\nabla_d (u_h-Q_hu)\|\\
& &+\|\Pi_h(bu)-bQ_0u\|\;\|\nabla_d( u_h-Q_hu)\|
+\|c(u-Q_0u)\|\;\|u_0-Q_0u\|.
\end{eqnarray*}
Next, we use the G{\aa}rding's inequality (\ref{garding}) to obtain
\begin{eqnarray*}
\alpha_1(\|\nabla_d(e_h)\|^2+\|e_{h,0}\|^2)&\le&
\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|\;\|\nabla_d (u_h-Q_hu)\|\\
& &+\|\Pi_h(bu)-bQ_0u\|\;\|\nabla_d( u_h-Q_hu)\|\\
& &+\|c(u-Q_0u)\|\;\|u_0-Q_0u\|+K\|u_0-Q_0u\|^2\\
&\le& \frac{\alpha_1}{2} (\|\nabla_d(u_h-Q_hu)\|^2+\|u_0-Q_0u\|^2)\\
& &+C\left(\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|^2 +
\|\Pi_h(bu)-bQ_0u\|^2\right.\\
& & +\left. \|c(u-Q_0u)\|^2\right)+K\|u_0-Q_0u\|^2,
\end{eqnarray*}
which implies the desired estimate (\ref{H1errorestimate}).
\end{proof}
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\subsection{An estimate in $L^2(\Omega)$}
We use the standard duality argument to derive an estimate for the
error $u_h-Q_h u$ in the standard $L^2$ norm over domain $\Omega$.
\begin{lemma}\label{l2-error}
Assume that the dual of the problem (\ref{pde}) and (\ref{bc}) has
the $H^{1+s}$ regularity. Let $u\in H^{1}(\Omega)$ be the solution
(\ref{pde}) and (\ref{bc}), and $u_h$ be a weak Galerkin
approximation of $u$ arising from (\ref{WG-fem}) by using either the
weak finite element space $S_h(j,j+1)$ or $S_h(j,j)$. Let $Q_hu$ be
the $L^2$ projection of $u$ in the corresponding finite element
space (recall that it is locally defined). Then, there exists a
constant $C$ such that
\begin{eqnarray*}
\|Q_0u-u_0\|&&\le Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& +\|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|\right),
\end{eqnarray*}
provided that the meshsize $h$ is sufficiently small.
\end{lemma}
\begin{proof}
Consider the dual problem of (\ref{pde}) and (\ref{bc}) which seeks
$w\in H_0^1(\Omega)$ satisfying
\begin{eqnarray}
-\nabla\cdot (a \nabla w)-b\cdot\nabla w+ cw &=& Q_0u-u_0\quad
\mbox{in}\;\Omega\label{dual}
\end{eqnarray}
The assumed $H^{1+s}$ regularity for the dual problem implies the
existence of a constant $C$ such that
\begin{equation}\label{reg}
\|w\|_{1+s}\le C\|Q_0u-u_0\|.
\end{equation}
Testing (\ref{dual}) against $Q_0u-u_0$ element by element gives
\begin{eqnarray}
\|Q_0u-u_0\|^2&=&(-\nabla\cdot (a \nabla w),\;Q_0u-u_0)-(b\cdot\nabla w,\;Q_0u-u_0)
+ (cw,\;Q_0u-u_0)\nonumber\\
&=&I+II+III,\label{m1}
\end{eqnarray}
where $I, II,$ and $III$ are defined to represent corresponding
terms. Let us estimate each of these terms one by one.
For the term $I$, we use the identity (\ref{4.200}) to obtain
\begin{eqnarray*}
I&=&(-\nabla\cdot (a\nabla w),Q_0u-u_0)= (\Pi_h (a\nabla
w),\nabla_d(Q_hu-u_h)).
\end{eqnarray*}
Recall that $\nabla_d(Q_hu)=R_h (\nabla u)$ with $R_h$ being a local
$L^2$ projection. Thus,
\begin{eqnarray}\nonumber
I&=& (\Pi_h (a\nabla w),\nabla_d(Q_hu-u_h))=(\Pi_h (a\nabla w),R_h \nabla u-\nabla_d u_h)\\
&=& (\Pi_h (a\nabla w),\nabla u-\nabla_d u_h)\nonumber\\
&=& (\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h) + (a\nabla
w,\nabla u-\nabla_d u_h).\label{yes.888}
\end{eqnarray}
The second term in the above equation above can be handled as
follows. Adding and subtracting two terms $(a\nabla_d Q_h w,
\nabla_d u_h)$ and $(a(\nabla w - R_h \nabla w), \nabla u)$ and
using the fact that $\nabla_d (Q_h u) =R_h (\nabla u)$ and the
definition of $R_h$, we arrive at
\begin{eqnarray}
(a\nabla w,\nabla u-\nabla_d u_h) &=& (a\nabla w, \nabla u)
-(a\nabla w, \nabla_d u_h) \nonumber\\
&=& (a\nabla w, \nabla u) -(a\nabla_d Q_h w, \nabla_d u_h)- (a(\nabla w - R_h \nabla w), \nabla_d u_h)\nonumber\\
&=& (a\nabla w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h)- (a(\nabla w - R_h \nabla w),
\nabla_d u_h - \nabla u)\nonumber\\
& & - (a(\nabla w - R_h \nabla w), \nabla u)\nonumber\\
&=& (a\nabla w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h) -
(a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\label{yes.889}\\
& & - (\nabla w - R_h \nabla w, a\nabla u -R_h (a\nabla
u)).\nonumber
\end{eqnarray}
Substituting (\ref{yes.889}) into (\ref{yes.888}) yields
\begin{eqnarray}\label{yes.termI}
I&=&(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h)- (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\\
& &-(\nabla w - R_h \nabla w, a\nabla u -R_h (a\nabla u))+(a\nabla
w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h).\nonumber
\end{eqnarray}
For the term $II$, we add and subtract $(\nabla_d(Q_h
w),\;b(Q_0u-u_0))$ from $II$ to obtain
\begin{eqnarray*}
II&=&-(b\cdot\nabla w,\;Q_0u-u_0)\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;b(Q_0u-u_0))\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;bQ_0u)+(\nabla_d(Q_h w),\;bu_0).
\end{eqnarray*}
In the following, we will deal with the second term on the right
hand side of the above equation. To this end, we use (\ref{4.88})
and the definition of $R_h$ and $Q_0$ to obtain
\begin{eqnarray*}
(\nabla_d(Q_h w),\;bQ_0u)&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u)+(\nabla w,\;bQ_0u)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(\nabla_d(Q_h w)-\nabla w,\;bu)\\
& &+(\nabla w,\;bQ_0u-bu)+(\nabla w,\;bu)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))\\
& &+(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)+(\nabla w,\;bu).
\end{eqnarray*}
Combining the last two equations above, we arrive at
\begin{eqnarray}\label{yes.termII}
II&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\\
& &-(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))-(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\nonumber\\
& &-(\nabla w,\;bu)+(\nabla_d(Q_h w),\;bu_0).\nonumber
\end{eqnarray}
As to the term $III$, by adding and subtracting some terms and
using the fact that $Q_0$ is a local $L^2$ projection, we easily
obtain the following
\begin{eqnarray*}
III&=& (cw,\;Q_0u-u_0)=(cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u-u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w-cw,\;Q_0u)+(cw,\;Q_0u-u)\\
& &+(cw,\;u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(Q_0w-w,\;cQ_0u-cu)+(Q_0w-w,\;cu-Q_0(cu))\\
& & +(cw-Q_0(cw),\;Q_0u-u)+(cw,\;u)-(cQ_0w,\;u_0).\\
\end{eqnarray*}
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For the term $II$, we add and subtract $(\nabla_d(Q_h
w),\;b(Q_0u-u_0))$ from $II$ to obtain
\begin{eqnarray*}
II&=&-(b\cdot\nabla w,\;Q_0u-u_0)\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;b(Q_0u-u_0))\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;bQ_0u)+(\nabla_d(Q_h w),\;bu_0).
\end{eqnarray*}
In the following, we will deal with the second term on the right
hand side of the above equation. To this end, we use (\ref{4.88})
and the definition of $R_h$ and $Q_0$ to obtain
\begin{eqnarray*}
(\nabla_d(Q_h w),\;bQ_0u)&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u)+(\nabla w,\;bQ_0u)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(\nabla_d(Q_h w)-\nabla w,\;bu)\\
& &+(\nabla w,\;bQ_0u-bu)+(\nabla w,\;bu)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))\\
& &+(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)+(\nabla w,\;bu).
\end{eqnarray*}
Combining the last two equations above, we arrive at
\begin{eqnarray}\label{yes.termII}
II&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\\
& &-(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))-(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\nonumber\\
& &-(\nabla w,\;bu)+(\nabla_d(Q_h w),\;bu_0).\nonumber
\end{eqnarray}
As to the term $III$, by adding and subtracting some terms and
using the fact that $Q_0$ is a local $L^2$ projection, we easily
obtain the following
\begin{eqnarray*}
III&=& (cw,\;Q_0u-u_0)=(cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u-u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w-cw,\;Q_0u)+(cw,\;Q_0u-u)\\
& &+(cw,\;u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(Q_0w-w,\;cQ_0u-cu)+(Q_0w-w,\;cu-Q_0(cu))\\
& & +(cw-Q_0(cw),\;Q_0u-u)+(cw,\;u)-(cQ_0w,\;u_0).\\
\end{eqnarray*}
Note that the sum of the last two terms in $I$ (see
(\ref{yes.termI})), $II$ (see (\ref{yes.termII})), and $III$ (see
the last equation above) gives
\begin{eqnarray*}
(a\nabla w, \nabla u)- &&(a\nabla_d Q_h w, \nabla_d u_h)-(\nabla
w,\;bu)+(\nabla_d(Q_h w),\;bu_0)
+(cw,\;u)-(cQ_0w,\;u_0)\\
&&=a(u,w)-a(u_h, Q_hw)\\
&&=(f,\; w)-(f,\; Q_0w)\\
&&=(f-Q_0f,\;w-Q_0w).
\end{eqnarray*}
Thus, the sum of $I$, $II$, and $III$ can be written as follows:
\begin{eqnarray}
\|Q_0u-u_0\|^2=&& (f-Q_0f,w-Q_0w)+(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h)\nonumber\\
&-& (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)- ((\nabla w - R_h \nabla w), a\nabla u -R_h (a\nabla u))\nonumber\\
&-&(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\nonumber\\
&-&(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))-(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\nonumber\\
&+&(cw-cQ_0w,\;Q_0u-u_0)+(Q_0w-w,\;cQ_0u-cu)\nonumber\\
&+&(Q_0w-w,\;cu-Q_0(cu))+(cw-Q_0(cw),\;Q_0u-u).\label{noname}
\end{eqnarray}
Using the triangle inequality, (\ref{4.88}) and (\ref{reg}), we can bound the second term
on the right hand side in the above equation by
\begin{eqnarray*}
\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d
u_h)\right|&\leq&
\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_dQ_h u)\right|\\
& & +\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla_dQ_h u-\nabla_d u_h)\right|\\
&\le&Ch^{s}\left(\|\nabla u-R_h(\nabla
u)\|+\|\nabla_d(Q_hu-u_h)\|\right)\|Q_0u-u_0\|.
\end{eqnarray*}
The other terms on the right hand side of (\ref{noname}) can be
estimated in a similar fashion, for which we state the results as
follows:
\begin{eqnarray*}
\left| (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\right|
&\le&Ch^{s}\left(\|\nabla u-R_h(\nabla
u)\|+\|\nabla_d(Q_hu-u_h)\|\right)\|Q_0u-u_0\|,\\
\left|((\nabla w - R_h \nabla w), a\nabla u -R_h (a\nabla u))\right|
&\le&Ch^{s}\|a\nabla u-R_h(a\nabla
u)\|\ \|Q_0u-u_0\|,\\
\left| (\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0)) \right|
&\le&Ch^{s} \|Q_0u-u_0\|^2,\\
\left|(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\right|
&\le&Ch^{s}\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left|(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))\right|
&\le&Ch^{s}\|bu-R_h(bu)\|\ \|Q_0u-u_0\|,\\
\left|(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\right|
&\le&Ch^{s}\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left| (cw-cQ_0w,\;Q_0u-u_0) \right|
&\le&Ch\|Q_0u-u_0\|^2,\\
\left|(Q_0w-w,\;cQ_0u-cu)\right|
&\le&Ch\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left|(Q_0w-w,\;cu-Q_0(cu))\right|
&\le&Ch\|cu-Q_0(cu)\|\ \|Q_0u-u_0\|,\\
\left|(cw-Q_0(cw),\;Q_0u-u)\right| &\le&Ch\|u-Q_0u\|\ \|Q_0u-u_0\|.
\end{eqnarray*}
Substituting the above estimates into (\ref{noname}) yields
\begin{eqnarray*}
\|Q_0u&&-u_0\|^2\le Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& + \|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|+\|Q_0u-u_0\|\right)\|Q_0u-u_0\|.
\end{eqnarray*}
For sufficiently small meshsize $h$, we have
\begin{eqnarray*}
\|Q_0u-u_0\|&\le & Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& +\|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|\right),
\end{eqnarray*}
which completes the proof.
\end{proof}
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\subsection{Error estimates in $H^1$ and $L^2$}
With the results established in Lemma \ref{h1-error} and Lemma
\ref{l2-error}, we are ready to derive an error estimate for the
weak Galerkin approximation $u_h$. To this end, we may substitute
the result of Lemma \ref{l2-error} into the estimate shown in Lemma
\ref{h1-error}. If so, for sufficiently small meshsize $h$, we would
obtain the following estimate:
\begin{eqnarray*}
\|\nabla_d(u_h-Q_hu)\|^2+\|u_0-Q_0u\|^2 &\le& C\left(\|\Pi_h(a\nabla
u)-aR_h(\nabla u)\|^2 +\|c(u-Q_0u)\|^2
\right.\\
&& +\left. \|\Pi_h(bu)-bQ_0u\|^2\right) \\
&& + Ch^{2s}\left(h^2\|f-Q_0f\|^2+\|\nabla u- R_h(\nabla
u)\|^2\right.\\
&& + \left. \|a\nabla u - R_h(a\nabla u)\|^2 +\|u-Q_0 u\|^2\right.\\
&& +\left. \|bu-R_h(bu)\|^2+ \|cu-Q_0(cu)\|^2\right).
\end{eqnarray*}
A further use of the interpolation error estimate leads to the
following error estimate in a discrete $H^1$ norm.
\begin{theorem}\label{H1error-estimate}
In addition to the assumption of Lemma \ref{l2-error}, assume that
the exact solution $u$ is sufficiently smooth such that $u\in
H^{m+1}(\Omega)$ with $0\le m \le j+1$. Then, there exists a
constant $C$ such that
\begin{eqnarray}\label{trueH1error}
\|\nabla_d(u_h-Q_hu)\|+\|u_0-Q_0u\| \le C(h^{m} \|u\|_{m+1}
+h^{1+s}\|f-Q_0f\|).
\end{eqnarray}
\end{theorem}
Now substituting the error estimate (\ref{trueH1error}) into the
estimate of Lemma \ref{l2-error}, and then using the standard
interpolation error estimate we obtain
\begin{eqnarray*}
\|u_h-Q_hu\|&\le& C\left(h^{1+s}\|f-Q_0f\|+h^{m+s} \|u\|_{m+1} +
h^s(h^{m} \|u\|_{m+1} +h^{1+s}\|f-Q_0f\|)\right)\\
&\le&C\left(h^{1+s}\|f-Q_0f\|+h^{m+s} \|u\|_{m+1}\right).
\end{eqnarray*}
The result can then be summarized as follows.
\begin{theorem}\label{trueL2error} Under the assumption of Theorem
\ref{H1error-estimate}, there exists a constant $C$ such that
\begin{eqnarray*}
\|u_h-Q_hu\| \le C\left(h^{1+s}\|f-Q_0f\|+h^{m+s}
\|u\|_{m+1}\right), \quad s\in (0,1],\ m\in (0,j+1],
\end{eqnarray*}
provided that the mesh-size $h$ is sufficiently small.
\end{theorem}
If the exact solution $u$ of (\ref{pde}) and (\ref{bc}) has the
$H^{j+2}$ regularity, then we have from Theorem \ref{trueL2error}
that
\begin{eqnarray*}
\|u_h-Q_hu\| &\le& C\left(h^{1+s}h^{j}\|f\|_{j}+h^{j+s+1}
\|u\|_{j+2}\right)\\
&\le& C h^{j+s+1} \left(\|f\|_{j}+\|u\|_{j+2}\right)
\end{eqnarray*}
for some $0<s\leq 1$, where $s$ is a regularity index for the dual
of (\ref{pde}) and (\ref{bc}). In the case that the dual has a full
$H^2$ (i.e., $s=1$) regularity, one would arrive at
\begin{eqnarray}\label{superc}
\|u_h-Q_hu\| \le C h^{j+2} \left(\|f\|_{j}+\|u\|_{j+2}\right).
\end{eqnarray}
Recall that on each triangular element $T^0$, the finite element
functions are of polynomials of order $j\ge 0$. Thus, the error
estimate (\ref{superc}) in fact reveals a superconvergence for the
weak Galerkin finite element approximation arising from
(\ref{WG-fem}).
\end{document}
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\begin{document}
\title{
Speeding-up $q$-gram mining on grammar-based compressed texts
}
\begin{abstract}
We present an efficient algorithm for calculating
$q$-gram frequencies on strings represented in compressed form,
namely, as a straight line program (SLP).
Given an SLP $\mathcal{T}$ of size $n$ that represents string $T$,
the algorithm computes the occurrence frequencies of
{\em all} $q$-grams in $T$, by reducing the problem to the weighted
$q$-gram frequencies problem on a trie-like structure of size
$m = |T|-\mathit{dup}(q,\mathcal{T})$,
where $\mathit{dup}(q,\mathcal{T})$ is a quantity
that represents the amount of redundancy that the SLP captures with respect to
$q$-grams.
The reduced problem can be solved in linear time.
Since $m = O(qn)$, the running time of our algorithm is
$O(\min\{|T|-\mathit{dup}(q,\mathcal{T}),qn\})$,
improving our previous $O(qn)$ algorithm when
$q = \Omega(|T|/n)$.
\end{abstract}
\section{Introduction}
Many large string data sets are usually first compressed and
stored, while they are decompressed afterwards in order to be used and
analyzed.
Compressed string processing (CSP) is an approach that has been
gaining attention in the string processing community.
Assuming that the input is given in compressed form,
the aim is to develop methods where the string is processed or analyzed without
explicitly decompressing the entire string,
leading to algorithms with time and space complexities
that depend on the compressed size rather than the
whole uncompressed size.
Since compression algorithms inherently capture regularities
of the original string, clever CSP algorithms
can be
theoretically~\cite{NJC97,crochemore03:_subquad_sequen_align_algor_unres_scorin_matric,hermelin09:_unified_algor_accel_edit_distan,gawrychowski11:_LZ_comp_str_fast_}, and
even
practically~\cite{shibata00:_speed_up_patter_match_text_compr,goto11:_fast_minin_slp_compr_strin},
faster than algorithms which process the uncompressed string.
In this paper, we assume that the input string
is represented as a Straight Line Program (SLP),
which is a context free grammar in Chomsky normal form
that derives a single string.
SLPs are a useful tool when considering CSP algorithms, since
it is known that outputs of various
grammar based compression algorithms~\cite{SEQUITUR,LarssonDCC99},
as well as dictionary compression algorithms~\cite{LZ78,LZW,LZ77,LZSS}
can be modeled efficiently by SLPs~\cite{rytter03:_applic_lempel_ziv}.
We consider the $q$-gram frequencies problem on compressed text
represented as SLPs.
$q$-gram frequencies have profound applications in the field of string
mining and classification.
The problem was first considered for the CSP setting
in~\cite{inenaga09:_findin_charac_subst_compr_texts},
where an $O(|\Sigma|^2n^2)$-time $O(n^2)$-space algorithm
for finding the {\em most frequent} $2$-gram from an
SLP of size $n$ representing text $T$ over alphabet $\Sigma$ was presented.
In~\cite{claudear:_self_index_gramm_based_compr}, it is claimed
that the most frequent $2$-gram can be found in $O(|\Sigma|^2n\log n)$-time
and $O(n\log|T|)$-space, if the SLP is pre-processed and a self-index is built.
A much simpler and efficient $O(qn)$
time and space algorithm for general $q \geq 2$ was recently
developed~\cite{goto11:_fast_minin_slp_compr_strin}.
Remarkably, computational experiments on various data sets
showed that the $O(qn)$ algorithm is actually faster than
calculations on uncompressed strings, when $q$ is
small~\cite{goto11:_fast_minin_slp_compr_strin}.
However, the algorithm
slows down considerably
compared to the uncompressed approach when $q$ increases.
This is because the algorithm reduces the
$q$-gram frequencies problem on an SLP of size $n$,
to the weighted $q$-gram frequencies problem on a weighted string of
size at most $2(q-1)n$.
As $q$ increases, the length of the string becomes longer than the
uncompressed string $T$.
Theoretically $q$ can be as large as $O(|T|)$,
hence in such a case the algorithm requires $O(|T|n)$ time,
which is worse than a trivial $O(|T|)$ solution that first decompresses the given SLP and
runs a linear time algorithm for $q$-gram frequencies computation on $T$.
In this paper, we solve this problem, and improve the previous $O(qn)$ algorithm
both theoretically and practically.
We introduce a $q$-gram neighbor relation on SLP variables,
in order to reduce the redundancy
in the partial decompression of the string which is performed in the
previous algorithm.
Based on this idea, we are able to convert the problem to a
weighted $q$-gram frequencies problem
on a weighted trie, whose size is at most
$|T|-\mathit{dup}(q,\mathcal{T})$.
Here, $\mathit{dup}(q,\mathcal{T})$ is a quantity that represents
the amount of redundancy that the SLP captures with respect to
$q$-grams. Since the size of the trie is also bounded by $O(qn)$,
the time complexity of our new algorithm is
$O(\min\{qn,|T|-\mathit{dup}(q,\mathcal{T})\})$,
improving on our previous $O(qn)$ algorithm when
$q = \Omega(|T|/n)$.
Preliminary computational experiments show that our new approach achieves
a practical speed up as well, for all values of $q$.
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\section{Introduction}
Many large string data sets are usually first compressed and
stored, while they are decompressed afterwards in order to be used and
analyzed.
Compressed string processing (CSP) is an approach that has been
gaining attention in the string processing community.
Assuming that the input is given in compressed form,
the aim is to develop methods where the string is processed or analyzed without
explicitly decompressing the entire string,
leading to algorithms with time and space complexities
that depend on the compressed size rather than the
whole uncompressed size.
Since compression algorithms inherently capture regularities
of the original string, clever CSP algorithms
can be
theoretically~\cite{NJC97,crochemore03:_subquad_sequen_align_algor_unres_scorin_matric,hermelin09:_unified_algor_accel_edit_distan,gawrychowski11:_LZ_comp_str_fast_}, and
even
practically~\cite{shibata00:_speed_up_patter_match_text_compr,goto11:_fast_minin_slp_compr_strin},
faster than algorithms which process the uncompressed string.
In this paper, we assume that the input string
is represented as a Straight Line Program (SLP),
which is a context free grammar in Chomsky normal form
that derives a single string.
SLPs are a useful tool when considering CSP algorithms, since
it is known that outputs of various
grammar based compression algorithms~\cite{SEQUITUR,LarssonDCC99},
as well as dictionary compression algorithms~\cite{LZ78,LZW,LZ77,LZSS}
can be modeled efficiently by SLPs~\cite{rytter03:_applic_lempel_ziv}.
We consider the $q$-gram frequencies problem on compressed text
represented as SLPs.
$q$-gram frequencies have profound applications in the field of string
mining and classification.
The problem was first considered for the CSP setting
in~\cite{inenaga09:_findin_charac_subst_compr_texts},
where an $O(|\Sigma|^2n^2)$-time $O(n^2)$-space algorithm
for finding the {\em most frequent} $2$-gram from an
SLP of size $n$ representing text $T$ over alphabet $\Sigma$ was presented.
In~\cite{claudear:_self_index_gramm_based_compr}, it is claimed
that the most frequent $2$-gram can be found in $O(|\Sigma|^2n\log n)$-time
and $O(n\log|T|)$-space, if the SLP is pre-processed and a self-index is built.
A much simpler and efficient $O(qn)$
time and space algorithm for general $q \geq 2$ was recently
developed~\cite{goto11:_fast_minin_slp_compr_strin}.
Remarkably, computational experiments on various data sets
showed that the $O(qn)$ algorithm is actually faster than
calculations on uncompressed strings, when $q$ is
small~\cite{goto11:_fast_minin_slp_compr_strin}.
However, the algorithm
slows down considerably
compared to the uncompressed approach when $q$ increases.
This is because the algorithm reduces the
$q$-gram frequencies problem on an SLP of size $n$,
to the weighted $q$-gram frequencies problem on a weighted string of
size at most $2(q-1)n$.
As $q$ increases, the length of the string becomes longer than the
uncompressed string $T$.
Theoretically $q$ can be as large as $O(|T|)$,
hence in such a case the algorithm requires $O(|T|n)$ time,
which is worse than a trivial $O(|T|)$ solution that first decompresses the given SLP and
runs a linear time algorithm for $q$-gram frequencies computation on $T$.
In this paper, we solve this problem, and improve the previous $O(qn)$ algorithm
both theoretically and practically.
We introduce a $q$-gram neighbor relation on SLP variables,
in order to reduce the redundancy
in the partial decompression of the string which is performed in the
previous algorithm.
Based on this idea, we are able to convert the problem to a
weighted $q$-gram frequencies problem
on a weighted trie, whose size is at most
$|T|-\mathit{dup}(q,\mathcal{T})$.
Here, $\mathit{dup}(q,\mathcal{T})$ is a quantity that represents
the amount of redundancy that the SLP captures with respect to
$q$-grams. Since the size of the trie is also bounded by $O(qn)$,
the time complexity of our new algorithm is
$O(\min\{qn,|T|-\mathit{dup}(q,\mathcal{T})\})$,
improving on our previous $O(qn)$ algorithm when
$q = \Omega(|T|/n)$.
Preliminary computational experiments show that our new approach achieves
a practical speed up as well, for all values of $q$.
\section{Preliminaries}
\subsection{Intervals, Strings, and Occurrences}
For integers $i \leq j$, let $[i:j]$ denote the interval of integers $\{i,\ldots, j\}$.
For an interval $[i:j]$ and integer $q > 0$,
let $\mathit{pre}([i:j],q)$ and $\mathit{suf}([i:j],q)$
represent respectively, the length-$q$ prefix and suffix interval,
that is,
$\mathit{pre}([i:j],q) = [i:\min(i+q-1,j)]$ and
$\mathit{suf}([i:j],q) = [\max(i,j-q+1):j]$.
Let $\Sigma$ be a finite {\em alphabet}.
An element of $\Sigma^*$ is called a {\em string}.
For any integer $q > 0$, an element of $\Sigma^q$ is called a \emph{$q$-gram}.
The length of a string $T$ is denoted by $|T|$.
The empty string $\varepsilon$ is a string of length 0,
namely, $|\varepsilon| = 0$.
For a string $T = XYZ$, $X$, $Y$ and $Z$ are called
a \emph{prefix}, \emph{substring}, and \emph{suffix} of $T$, respectively.
The $i$-th character of a string $T$ is denoted by $T[i]$, where $1 \leq i \leq |T|$.
For a string $T$ and interval $[i:j] (1 \leq i \leq j \leq |T|)$, let $T([i:j])$
denote the substring of $T$ that begins at position $i$ and ends at
position $j$.
For convenience, let $T([i:j]) = \varepsilon$ if $j < i$.
For a string $T$ and integer $q \geq 0$,
let $\mathit{pre}(T,q)$ and $\mathit{suf}(T,q)$
represent respectively, the length-$q$ prefix and suffix of $T$,
that is,
$\mathit{pre}(T,q) =T(\mathit{pre}([1:|T|],q))$ and $\mathit{suf}(T,q) = T(\mathit{suf}([1:|T|],q))$.
For any strings $T$ and $P$,
let $\mathit{Occ}(T,P)$ be the set of occurrences of $P$ in $T$, i.e.,
$\mathit{Occ}(T,P) = \{k > 0 \mid T[k:k+|P|-1] = P\}$.
The number of elements $|\mathit{Occ}(T,P)|$ is called
the \emph{occurrence frequency} of $P$ in $T$.
\subsection{Straight Line Programs}
\begin{wrapfigure}[11]{r}{0.5\textwidth}
\centerline{\includegraphics[width=0.45\textwidth]{slp.eps}}
\caption{
The derivation tree of
SLP $\mathcal T = \{ X_1 \rightarrow \mathtt{a}$, $X_2 \rightarrow \mathtt{b}$, $X_3 \rightarrow X_1X_2$,
$X_4 \rightarrow X_1X_3$, $X_5 \rightarrow X_3X_4$, $X_6
\rightarrow X_4X_5$, $X_7 \rightarrow X_6X_5 \}$,
representing string $T = \mathit{val}(X_7) = \mathtt{aababaababaab}$.
}
\label{fig:SLP}
\end{wrapfigure}
A {\em straight line program} ({\em SLP}) is a set of assignments
$\mathcal T = \{ X_1 \rightarrow expr_1, X_2 \rightarrow expr_2, \ldots, X_n \rightarrow expr_n\}$,
where each $X_i$ is a variable and each $expr_i$ is an expression, where
$expr_i = a$ ($a\in\Sigma$), or $expr_i = X_{\ell(i)} X_{r(i)}$~($i > \ell(i),r(i)$).
It is essentially a context free grammar in the Chomsky normal form, that derives a single string.
Let $\mathit{val}(X_i)$ represent the string derived from variable $X_i$.
To ease notation, we sometimes associate $\mathit{val}(X_i)$ with $X_i$ and
denote $|\mathit{val}(X_i)|$ as $|X_i|$,
and $\mathit{val}(X_i)([u:v])$ as $X_i([u:v])$ for any interval $[u:v]$.
An SLP $\mathcal{T}$ {\em represents} the string $T = \mathit{val}(X_n)$.
The \emph{size} of the program $\mathcal T$ is the number $n$ of
assignments in $\mathcal T$.
Note that $|T|$ can be as large as $\Theta(2^n)$. However, we assume
as in various previous work on SLP,
that the computer word size is at least $\log |T|$, and hence,
values representing lengths and positions of $T$
in our algorithms can be manipulated in constant time.
The derivation tree of SLP $\mathcal{T}$ is a labeled
ordered binary tree where each internal node is labeled with a
non-terminal variable in $\{X_1,\ldots,X_n\}$, and each leaf is labeled with a terminal character in $\Sigma$.
The root node has label $X_n$.
Let $\mathcal{V}$ denote the set of internal nodes in
the derivation tree.
For any internal node $v\in\mathcal{V}$,
let $\langle v\rangle$ denote the index of its label
$\variable{v}$.
Node $v$ has a single child which is a leaf labeled with $c$
when $(\variable{v} \rightarrow c) \in \mathcal{T}$ for some $c\in\Sigma$,
or
$v$ has a left-child and right-child respectively denoted $\ell(v)$ and $r(v)$,
when
$(\variable{v}\rightarrow \variable{\ell(v)}\variable{r(v)}) \in \mathcal{T}$.
Each node $v$ of the tree derives $\mathit{val}(\variable{v})$,
a substring of $T$,
whose corresponding interval $\mathit{val}Int(v)$,
with $T(\mathit{val}Int(v)) = \mathit{val}(\variable{v})$,
can be defined recursively as follows.
If $v$ is the root node, then $\mathit{val}Int(v) = [1:|T|]$.
Otherwise, if $(\variable{v}\rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{T}$,
then,
$\mathit{val}Int(\ell(v)) = [b_v:b_v+|\variable{\ell(v)}|-1]$
and
$\mathit{val}Int(r(v)) = [b_v+|\variable{\ell(v)}|:e_v]$,
where $[b_v:e_v] = \mathit{val}Int(v)$.
Let $\mathit{vOcc}(X_i)$ denote the number of times a variable $X_i$ occurs
in the derivation tree, i.e.,
$\mathit{vOcc}(X_i) = |\{ v \mid \variable{v}=X_i\}|$.
We assume that any variable $X_i$ is used at least once,
that is $\mathit{vOcc}(X_i) > 0$.
For any interval $[b:e]$ of $T (1\leq b \leq e \leq |T|)$,
let $\xi_\mathcal{T}(b,e)$ denote the deepest node $v$ in the derivation tree,
which derives an interval containing $[b:e]$, that is,
$\mathit{val}Int(v)\supseteq [b:e]$,
and no proper descendant of $v$
satisfies this condition.
We say that node $v$ {\em stabs} interval $[b:e]$,
and $\variable{v}$ is called the variable that stabs the interval.
If $b = e$, we have that
$(\variable{v} \rightarrow c) \in \mathcal{T}$ for some $c\in\Sigma$,
and $\mathit{val}Int(v) = b = e$.
If $b < e$, then
we have $(\variable{v} \rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{T}$,
$b\in \mathit{val}Int(\ell(v))$, and $e\in\mathit{val}Int(r(v))$.
When it is not confusing, we will sometimes
use $\xi_\mathcal{T}(b,e)$ to denote the variable $\variable{\xi_\mathcal{T}(b,e)}$.
SLPs can be efficiently pre-processed to hold various information.
$|X_i|$ and $\mathit{vOcc}(X_i)$ can be computed for all variables $X_i
(1\leq i\leq n)$ in a total of $O(n)$ time by a simple dynamic
programming algorithm.
Also, the following Lemma is useful for partial decompression of
a prefix of a variable.
\begin{lemma}[\cite{gasieniec05:_real_time_traver_gramm_based_compr_files}]
\label{label:prefix_decompression}
Given an SLP $\mathcal{T} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$,
it is possible to pre-process $\mathcal{T}$ in $O(n)$ time and
space, so that for any variable $X_i$ and $1 \leq j \leq |X_i|$,
${X_i}([1:j])$ can be computed in $O(j)$ time.
\end{lemma}
The formal statement of the problem we solve is:
\begin{problem}[$q$-gram frequencies on SLP]
\label{problem:SLPqgramfreq}
Given integer $q\geq 1$ and an SLP $\mathcal{T}$ of size $n$ that represents string $T$,
output $(i, |\mathit{Occ}(T,P)|)$ for all $P\in\Sigma^q$ where
$\mathit{Occ}(T,P)\neq\emptyset$, and some $i\in\mathit{Occ}(T,P)$.
\end{problem}
Since the problem is very simple for $q = 1$,
we shall only consider the case for $q\geq 2$ for the rest of the paper.
Note that although the number of distinct $q$-grams in $T$ is bounded by $O(qn)$,
we would require an extra multiplicative $O(q)$ factor for the output if we output
each $q$-gram explicitly as a string.
In our algorithms to follow, we compute
a compact, $O(qn)$-size representation of the output,
from which each $q$-gram can be easily obtained in $O(q)$ time.
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\section{$O(qn)$ Algorithm~\cite{goto11:_fast_minin_slp_compr_strin}}
\label{section:qn}
In this section, we briefly describe the $O(qn)$ algorithm
presented in~\cite{goto11:_fast_minin_slp_compr_strin}.
The idea is to count occurrences of $q$-grams with respect to
the variable that stabs its occurrence.
The algorithm reduces Problem~\ref{problem:SLPqgramfreq}
to calculating the frequencies of all $q$-grams in a weighted set of strings,
whose total length is $O(qn)$.
Lemma~\ref{lemma:qn_key}
shows the key idea of the algorithm.
\begin{lemma}
\label{lemma:qn_key}
For any SLP $\mathcal{T} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$
that represents string $T$, integer $q \geq 2$, and $P \in \Sigma^q$,
$|\mathit{Occ}(T,P)| = \sum_{i=1}^n\mathit{vOcc}(X_i)\cdot |\mathit{Occ}(t_i, P)|$,
where
$t_i = \mathit{suf}(\mathit{val}(X_{\ell(i)}),q-1)\mathit{pre}(\mathit{val}(X_{r(i)}),q-1)$.
\end{lemma}
\begin{proof}
For any $q \geq 2$, $v$ stabs the interval $[u:u+q-1]$
if and only if
$[u:u+q-1]\subseteq
[s_v:f_v] =
\mathit{suf}(\mathit{val}Int(\ell(v)),q-1)\cup\mathit{pre}(\mathit{val}Int(r(v)),q-1)$.
(See Fig.~\ref{fig:SLP-kgram}.)
Also, since an occurrence of $X_i$ in the derivation tree always
derives the same string $\mathit{val}(X_i)$,
$t_i = T([s_v:f_v])$ for any node $v$ such that $\variable{v} = X_i$.
Therefore,
\begin{eqnarray*}
\lefteqn{|\mathit{Occ}(T,P)| = \big|\{ u>0 \mid T([u:u+q-1]) = P\}\big|}\\
& = & \sum_{v\in \mathcal{V}} \big| \{ u>0 \mid \xi_{\mathcal{T}}(u,u+q-1)=v, j=u-s_v+1, \variable{v}([j:j+q-1]) = P \}\big| \\
& = & \sum_{i=1}^n \sum_{v\in \mathcal{V}: \variable{v}=X_i} \big|\{
u>0 \mid \xi_{\mathcal{T}}(u,u+q-1)=v,j=u-s_v+1, \variable{v}([j:j+q-1]) = P\}\big|\\
& = & \sum_{i=1}^n \sum_{v\in \mathcal{V}: \variable{v}=X_i} \mathit{Occ}(T([s_v:f_v]),P)
= \sum_{i=1}^n \mathit{vOcc}(X_i)\cdot \mathit{Occ}(t_i,P).\\
\end{eqnarray*}
\qed
\end{proof}
\begin{wrapfigure}[15]{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.4\textwidth]{SLP_ngram.eps}
\end{center}
\caption{
Length-$q$ intervals where
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$,
and $(X_i\rightarrow X_{\ell(i)} X_{r(i)}) \in \mathcal{T}$.
}
\label{fig:SLP-kgram}
\end{wrapfigure}
From Lemma~\ref{lemma:qn_key}, we have that occurrence frequencies in
$T$ are equivalent to occurrence frequencies in $t_i$ weighted by $\mathit{vOcc}(X_i)$.
Therefore, the $q$-gram frequencies problem can be regarded as
obtaining the {\em weighted} frequencies of all $q$-grams in the set
of strings $\{t_1,\ldots,t_n\}$,
where each occurrence of a $q$-gram in $t_i$ is weighted by $\mathit{vOcc}(X_i)$.
This can be further reduced to a weighted $q$-gram frequency
problem for a single string $z$, where each position of $z$
holds a weight associated with the $q$-gram that starts at that position.
String $z$ is constructed by concatenating all $t_i$'s with length at
least $q$.
The weights of positions corresponding to the first $|t_i| - (q-1)$
characters of $t_i$ will be $\mathit{vOcc}(X_i)$, while the last $(q-1)$
positions will be $0$ so that superfluous $q$-grams generated by the
concatenation are not counted.
The remaining is a simple linear time algorithm
using suffix and lcp arrays on the weighted string,
thus solving the problem in $O(qn)$ time and space.
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\section{New Algorithm}
\label{section:new_algorithm}
We now describe our new algorithm which solves the $q$-gram
frequencies problem on SLPs.
The new algorithm basically follows the previous $O(qn)$ algorithm,
but is an elegant refinement.
The reduction for the previous $O(qn)$ algorithm
leads to a fairly large amount of redundantly decompressed regions of
the text as $q$ increases.
This is due to the fact that
the $t_i$'s are considered independently for each variable $X_i$,
while {\em neighboring} $q$-grams that are stabbed by different
variables actually share $q-1$ characters.
The key idea of our new algorithm is to exploit this redundancy.
(See Fig.~\ref{fig:qgramneighbor}.)
In what follows, we introduce the concept of $q$-gram neighbors,
and reduce the $q$-gram frequencies problem on SLP to
a weighted $q$-gram frequencies problem on a weighted tree.
\subsection{$q$-gram Neighbor Graph}
We say that
$X_j$ is a {\em right $q$-gram neighbor} of $X_i$ $(i \neq j)$,
or equivalently,
$X_i$ is a {\em left $q$-gram neighbor} of $X_j$,
if for some integer $u \in [1:|T|-q]$,
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$ and
$\variable{\xi_\mathcal{T}(u+1,u+q)} = X_j$.
Notice that $|X_i|$ and $|X_j|$ are both at least $q$
if $X_i$ and $X_j$ are right or left $q$-gram neighbors of each other.
\begin{figure}
\caption{$q$-gram neighbors and redundancies.
(Left) $X_j$ is a right $q$-gram neighbor of
$X_i$, and $X_i$ is {\em a}
\label{fig:qgramneighbor}
\end{figure}
\begin{definition}
For $q\geq 2$, the right $q$-gram neighbor graph of SLP
$\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$ is the directed graph
$G_q = (V,E_r)$, where
\begin{eqnarray*}
V&=&\{ X_i \mid i \in \{1,\ldots, n\}, |X_i| \geq q \}\\
E_r &=& \{ (X_i,X_j) \mid X_j \mbox{ is a right $q$-gram neighbor of
$X_i$ } \}
\end{eqnarray*}
\end{definition}
Note that there can be multiple right $q$-gram neighbors for a given variable.
However,
the total number of edges in the neighbor graph
is bounded by $2n$, as will be shown below.
\begin{lemma}
\label{lemma:unique_neighbors}
Let $X_j$ be a right $q$-gram neighbor of $X_i$.
If, $|X_{r(i)}| \geq q$, then $X_j$ is the label of the deepest
variable on the left-most path of the derivation tree rooted at
a node labeled $X_{r(i)}$ whose length is at least $q$.
Otherwise, if $|X_{r(i)}| < q$, then
$X_i$ is the label of the deepest variable on the right-most path rooted at
a node labeled $X_{\ell(j)}$
whose length is at least $q$.
\end{lemma}
\begin{proof}
Suppose $|X_{r(i)}| \geq q$.
Let $u$ be a position, where
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$ and $\variable{\xi_\mathcal{T}(u+1,u+q)} = X_j$.
Then, since the interval $[u+1:u+q]$ is a prefix of
$\mathit{val}Int(X_{r(i)})$,
$X_j$ must be on the left most path rooted at $X_{r(i)}$.
Since $X_j = \variable{\xi_\mathcal{T}(u+1,u+q)}$,
the lemma follows from the definition of $\xi_\mathcal{T}$.
The case for $|X_{r(i)}| < q$ is symmetrical
and can be shown similarly.
\qed
\end{proof}
\begin{lemma}
For an arbitrary SLP
$\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$
and integer $q\geq 2$,
the number of edges in the right $q$-gram neighbor graph $G_q$ of $\mathcal{T}$ is at most $2n$.
\end{lemma}
\begin{proof}
Suppose $X_j$ is a right $q$-gram neighbor of $X_i$.
From Lemma~\ref{lemma:unique_neighbors}, we have that
if $|X_{r(i)}| \geq q$, the right $q$-gram neighbor of $X_i$ is uniquely determined
and that $|X_{\ell(j)}| < q$.
Similarly, if $|X_{r(i)}| < q$, $|X_{\ell(j)}|\geq q$ and the left $q$-gram neighbor of $X_j$
is uniquely $X_i$.
Therefore,
\begin{eqnarray*}
&&\sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| \geq q \}|
+ \sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| < q \}|\\
&=&
\sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| \geq q \}|
+ \sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{\ell(j)}| \geq q \}| \leq 2n.
\end{eqnarray*}
\qed
\end{proof}
\begin{lemma}
For an arbitrary SLP $\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$ and
integer $q\geq 2$,
the right $q$-gram neighbor graph $G_q$ of $\mathcal{T}$ can be constructed in $O(n)$ time.
\end{lemma}
\begin{proof}
For any variable $X_i$, let
$\mathit{lm}_q(X_i)$ and $\mathit{rm}_q(X_i)$ respectively represent the
index of the label of the deepest
node with length at least $q$
on the left-most and right-most path
in the derivation tree rooted at $X_i$, or $\mathit{null}$ if $|X_i| < q$.
These values can be computed for all variables in
a total of $O(n)$ time based on the following recursion:
If $(X_i \rightarrow a)\in\mathcal{T}$ for some $a\in\Sigma$, then
$\mathit{lm}_q(X_i) = \mathit{rm}_q(X_i) = \mathit{null}$.
For $(X_i \rightarrow X_{\ell(i)}X_{r(i)})\in\mathcal{T}$,
\begin{equation*}
\mathit{lm}_q(X_i) = \begin{cases}
\mathit{null} & \mbox{if } |X_i| < q,\\
i & \mbox{if } |X_i| \geq q \mbox{ and } |X_{\ell(i)}| < q,\\
\mathit{lm}_q(X_{\ell(i)}) & \mbox{otherwise. }
\end{cases}
\end{equation*}
$\mathit{rm}_q(X_i)$ can be computed similarly.
Finally,
\begin{eqnarray*}
E_r & = &
\{ (X_i, X_{\mathit{lm}_q(X_{r(i)})}) \mid \mathit{lm}_q(X_{r(i)}) \neq \mathit{null},
i = 1,\ldots, n \}\\
&&\cup \{ (X_{\mathit{rm}_q(X_{\ell(i)})}, X_i) \mid \mathit{rm}_q(X_{\ell(i)}) \neq
\mathit{null},
i=1,\ldots, n\}.
\end{eqnarray*}
\qed
\end{proof}
\begin{lemma}
\label{lemma:connected}
Let $G_q = (V, E_r)$ be the right $q$-gram neighbor graph of
SLP $\mathcal{T} = \{ X_i = expr_i \}_{i=1}^n$ representing string $T$,
and let $X_{i_1} = \variable{\xi_\mathcal{T}(1,q)}$.
Any variable $X_j \in V (i_1 \neq j)$ is reachable from $X_{i_1}$,
that is, there exists a directed path from $X_{i_1}$ to $X_j$ in $G_q$.
\end{lemma}
\begin{proof}
Straightforward, since any $q$-gram of $T$ except for the left most
$T([1:q])$ has a $q$-gram on its left.\qed
\end{proof}
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\subsection{Weighted $q$-gram Frequencies Over a Trie}
\label{section:weighted_q-gram_frequencies_over_a_trie}
From Lemma~\ref{lemma:connected}, we have that the right $q$-gram
neighbor graph is connected.
Consider an arbitrary directed spanning tree
rooted at $X_{i_1} = \variable{\xi_\mathcal{T}(1,q)}$ which can be obtained in linear time by a depth
first traversal on $G_q$ from $X_{i_1}$.
We define the label $\mathit{label}(X_i)$ of each node $X_i$ of the
$q$-gram neighbor graph, by \[\mathit{label}(X_i) = t_i[q:|t_i|] \]
where $t_i =
\mathit{suf}(\mathit{val}(X_{\ell(i)}),q-1)\mathit{pre}(\mathit{val}(X_{r(i)}),q-1)$ as before.
For convenience, let $X_{i_0}$ be a dummy variable
such that
$\mathit{label}(X_{i_0}) = T([1:q-1])$, and
$X_{r(i_0)} = X_{i_1}$ (and so $(X_{i_0},X_{i_1})\in E_r$).
\begin{lemma}
\label{lemma:path}
Fix a directed spanning tree on the right $q$-gram neighbor graph
of SLP $\mathcal{T}$, rooted at $X_{i_0}$.
Consider a directed path $X_{i_0}, \ldots, X_{i_m}$ on the spanning tree.
The weighted $q$-gram frequencies on the string obtained by
the concatenation $\mathit{label}(X_{i_0}) \mathit{label}(X_{i_1}) \cdots \mathit{label}(X_{i_m})$,
where each occurrence of a $q$-gram that ends in a position in
$\mathit{label}(X_{i_j})$ is weighted by
$\mathit{vOcc}(X_{i_j})$,
is equivalent to the
weighted $q$-gram frequencies of
strings $\{t_{i_1}, \ldots t_{i_m}\}$
where each $q$-gram in $t_{i_j}$ is weighted by $\mathit{vOcc}(X_{i_j})$.
\end{lemma}
\begin{proof}
Proof by induction:
for $m = 1$,
we have that $\mathit{label}(X_{i_0})\mathit{label}(X_{i_1}) = t_{i_1}$.
All $q$-grams in $t_{i_1}$ end in $t_{i_1}$
and so are weighted by $\mathit{vOcc}(X_{i_1})$.
When $\mathit{label}(X_{i_j})$ is added to
$\mathit{label}(X_{i_0}) \cdots \mathit{label}(X_{i_{j-1}})$,
$|\mathit{label}(X_{i_j})|$ new $q$-grams are formed, which correspond to
$q$-grams in $t_{i_j}$, i.e. $|t_{i_j}| = q - 1 + |\mathit{label}(X_{i_j})|$,
and $t_{i_j}$ is a suffix of $\mathit{label}(X_{i_{j-1}})\mathit{label}(X_{i_{j}})$.
All the new $q$-grams end in $\mathit{label}(X_{i_j})$ and are thus
weighted by $\mathit{vOcc}(X_{i_j})$.
\qed
\end{proof}
From Lemma~\ref{lemma:path}, we can construct a weighted trie $\Upsilon$
based on a directed spanning tree of $G_q$ and $\mathit{label}()$,
where
the weighted $q$-grams in $\Upsilon$ (represented as length-$q$ paths)
correspond to the
occurrence frequencies of $q$-grams in $T$.
\footnote{A minor technicality is that a node in $\Upsilon$ may have
multiple children with the same character label, but this
does not affect the time complexities of the algorithm.}
\begin{algorithm2e}[t]
\caption{Constructing weighted trie from SLP}
\label{algo:slp2trie}
Construct right $q$-gram neighbor graph $G=(V,E_r)$\;
Calculate $\mathit{vOcc}(X_i)$ for $i = 1,\ldots, n$\;
Calculate $|\mathit{label}(X_i)|$ for $i = 1,\ldots, n$\;
\lFor{$i = 0,\ldots, n$}{
$\mathsf{visited}[i] = \mathsf{false}$\;
}
$X_{i_1} = \variable{\xi_\mathcal{T}(1,q)} = \mathit{lm}_q(X_n)$\;
Define $X_{i_0}$ so that $X_{r(i_0)} = X_{i_1}$ and $|\mathit{label}(X_{i_0})| = q-1$\;
$\mathit{root} \leftarrow$ new node\tcp*[l]{root of resulting trie}
\ref{procedure:bdf}($i_0$, $\mathit{root}$)\;
\Return $\mathit{root}$
\end{algorithm2e}
\begin{procedure}[t]
\caption{BuildDepthFirst($i$, $\mathit{trieNode}$)}
\label{procedure:bdf}
\SetKw{KwAND}{and}
\SetKw{KwOR}{or}
\SetKw{KwBREAK}{break}
\SetKwFunction{BDF}{\ref{procedure:bdf}}
\tcp{add prefix of $r(i)$ to trieNode while right neighbors of $i$ are unique}
$l \leftarrow 0$; $k \leftarrow i$\;
\While{
{$\textsf{true}$}
}{
$l \leftarrow l + |\mathit{label}(X_k)|$\;
$\mathsf{visited}[k] \leftarrow \mathsf{true}$\;
\tcp{exit loop if right neighbor is possibly non-unique or is visited}
\lIf{$|X_{r(k)}| < q$ \KwOR $\mathsf{visited}[\mathit{lm}_q(X_{r(k)})] =
\mathsf{true}$}{
{\KwBREAK}\;
}
$k \leftarrow\mathit{lm}_q(X_{r(k)})$\;
}
add new branch from $\mathit{trieNode}$ with string $X_{r(i)}([1:l])$\;\label{algo:prefadd}
let end of new branch be $\mathit{newTrieNode}$\;
\tcp{If $|X_{r(k)}| < q$, there may be multiple right neighbors.}
\tcp{If $|X_{r(k)}|\geq q$, nothing is done because it has already been visited.}
\For{$X_c \in \{ X_j \mid (X_k,X_j) \in E_r \}$}{
\If{$\mathsf{visited}[c] = \mathsf{false}$}{
\BDF($X_c$, $\mathit{newTrieNode}$)\;
}
}
\end{procedure}
\begin{lemma}
$\Upsilon$ can be constructed in time linear in its size.
\end{lemma}
\begin{proof}
See Algorithm~\ref{algo:slp2trie}.
Let $G$ be the $q$-gram neighbor graph.
We construct $\Upsilon$ in a depth first manner starting at $X_{i_0}$.
The crux of the algorithm is that rather than
computing $\mathit{label}()$ separately for each variable,
we are able to aggregate the $\mathit{label}()$s and limit all partial
decompressions of variables to prefixes of variables, so that
Lemma~\ref{label:prefix_decompression} can be used.
Any directed acyclic path on $G$ starting at $X_{i_0}$
can be segmented into multiple sequences of variables,
where each sequence $X_{i_j}, \ldots, X_{i_k}$
is such that $j$ is the only integer in $[j:k]$ such that
$j = 0 $ or $|X_{r(i_{j-1})}| < q$.
From Lemma~\ref{lemma:unique_neighbors},
we have that $X_{i_{j+1}},\ldots,X_{i_k}$
are uniquely determined.
If $j>0$, $\mathit{label}(X_{i_j})$ is a prefix of $\mathit{val}(X_{r(i_j)})$
since $|X_{r(i_{j-1})}| < q$
(see Fig.~\ref{fig:qgramneighbor} Right),
and if $j=0$, $\mathit{label}(X_{i_0})$ is again a prefix of
$\mathit{val}(X_{r(i_0)}) = \mathit{val}(X_{i_1})$.
It is not difficult to see that $\mathit{label}(X_{i_j})\cdots\mathit{label}(X_{i_{k}})$ is
also a prefix of $X_{r(i_j)}$ since
$X_{i_{j+1}},\ldots,X_{i_k}$ are all descendants of $X_{r(i_j)}$, and
each $\mathit{label}()$ extends the
partially decompressed string to consider consecutive $q$-grams in $X_{r(i_j)}$.
Since prefixes of variables of SLPs can be decompressed in time
proportional to the output size with linear time pre-processing
(Lemma~\ref{label:prefix_decompression}), the lemma follows.
\qed
\end{proof}
We only illustrate how the character labels are determined
in the pseudo-code of Algorithm~\ref{algo:slp2trie}.
It is straightforward to assign a weight $\mathit{vOcc}(X_k)$ to
each node of $\Upsilon$ that corresponds to $\mathit{label}(X_k)$.
\begin{lemma}
\label{lemma:size_of_trie}
The number of edges in $\Upsilon$ is
$(q-1) + \sum \{ |t_i|-(q-1) \mid |X_i| \geq q, i=1,\ldots,n\} =
|T| - \mathit{dup}(q,\mathcal{T})$ where
\[
\mathit{dup}(q,\mathcal{T})
= \sum\{ (\mathit{vOcc}(X_i) - 1) \cdot (|t_i| - (q-1)) \mid |X_i| \geq q, i=1,\ldots,n \}\}
\]
\end{lemma}
\begin{proof}
$(q-1)+\sum \{ |t_i|-(q-1) \mid |X_i| \geq q, i=1,\ldots,n\}$ is straight
forward from the definition of $\mathit{label}(X_i)$ and the
construction of $\Upsilon$.
Concerning $\mathit{dup}$,
each variable $X_i$ occurs $\mathit{vOcc}(X_i)$ times
in the derivation tree, but only once in the directed spanning tree.
This means that for each occurrence after the first,
the size of $\Upsilon$ is reduced by $|\mathit{label}(X_i)|=|t_i| - (q-1)$ compared to $T$.
Therefore, the lemma follows.
\qed
\end{proof}
To efficiently count the weighted $q$-gram frequencies on $\Upsilon$,
we can use suffix trees.
A suffix tree for a trie is defined as a generalized suffix tree
for the set of strings represented in the trie as leaf to root paths.
\footnote{
When considering leaf to root paths on $\Upsilon$,
the direction of the string is the reverse of what is
in $T$. However, this is merely a matter of representation of the
output.
}
The following is known.
\begin{lemma}[\cite{shibuya03:_const_suffix_tree_tree_large_alphab}]
Given a trie of size $m$,
the suffix tree for the trie can be constructed in
$O(m)$ time and space.
\end{lemma}
With a suffix tree, it is a simple exercise to solve the weighted
$q$-gram frequencies problem on $\Upsilon$ in linear time.
In fact, it is known that the suffix array for the common suffix trie
can also be constructed in
linear time~\cite{ferragina09:_compr}, as well as its longest common prefix
array~\cite{kimura11}, which can also be used to solve the problem in
linear time.
\begin{corollary}
The weighted $q$-gram frequencies problem on a trie of size $m$
can be solved in $O(m)$ time and space.
\end{corollary}
From the above arguments, the theorem follows.
\begin{theorem}
The $q$-gram frequencies problem on an SLP $\mathcal{T}$ of size
$n$, representing string $T$ can be solved in
$O(\min\{qn,|T| - \mathit{dup}(q,\mathcal{T})\})$ time and space.
\end{theorem}
Note that since each $q\leq |t_i|\leq 2(q-1)$,
and $|\mathit{label}(X_i)| = |t_i| - (q-1)$, the
total length of decompressions made by the algorithm, i.e. the size
of the reduced problem, is at least halved and can be as small as $1/q$
(when all $|t_i|=q$, for example, in an SLP that represents LZ78
compression),
compared to the previous $O(qn)$ algorithm.
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\section{Preliminary Experiments}
We first evaluate the size of the trie $\Upsilon$ induced from the
right $q$-gram neighbor graph, on which the running time of the new algorithm
of Section~\ref{section:new_algorithm} is dependent.
We used data sets obtained from Pizza \& Chili Corpus,
and constructed SLPs using the RE-PAIR~\cite{LarssonDCC99} compression algorithm.
Each data is of size 200MB.
Table~\ref{table:zsize} shows the sizes of $\Upsilon$ for different values of $q$,
in comparison with the total length of strings $t_i$,
on which the previous $O(qn)$-time algorithm of Section~\ref{section:qn} works.
We cumulated the lengths of all $t_i$'s only for those satisfying $|t_i| \geq q$,
since no $q$-gram can occur in $t_i$'s with $|t_i| < q$.
Observe that for all values of $q$ and for all data sets,
the size of $\Upsilon$ (i.e., the total number of characters in $\Upsilon$) is
smaller than those of $t_i$'s and the original string.
\begin{table}[t]
\caption{
A comparison of the size of $\Upsilon$ and the total length of strings $t_i$ for
SLPs that represent textual data from Pizza \& Chili Corpus.
The length of the original text is 209,715,200.
The SLPs were constructed by RE-PAIR~\cite{LarssonDCC99}.
}
\label{table:zsize}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{1pt}
\renewcommand{ptm}{ptm}
\renewcommand{phv}{phv}
\renewcommand{pcr}{pcr}
\normalfont
\input{table_zsize.tex}
\end{center}
\end{table}
The construction of the suffix tree or array for a trie,
as well as the algorithm for Lemma~\ref{label:prefix_decompression},
require various tools such as level ancestor
queries~\cite{dietz91:_findin,berkman94:_findin,bender04:_level_ances_probl}
for which we did not have an efficient implementation.
Therefore, we try to assess the practical impact of the reduced problem size
using a simplified version of our new algorithm.
We compared three algorithms ($\textrm{NSA}$, $\textrm{SSA}$, $\textrm{STSA}$) that count the occurrence frequencies of all
$q$-grams in a text given as an SLP.
$\textrm{NSA}$ is the $O(|T|)$-time algorithm which works on the uncompressed text,
using suffix and LCP arrays.
$\textrm{SSA}$ is our previous $O(qn)$-time algorithm~\cite{goto11:_fast_minin_slp_compr_strin},
and $\textrm{STSA}$ is a simplified version of our new algorithm.
$\textrm{STSA}$ further reduces the weighted $q$-gram frequencies problem on $\Upsilon$,
to a weighted $q$-gram frequencies problem on a single string as follows:
instead of constructing $\Upsilon$, each branch of $\Upsilon$ (on line~\ref{algo:prefadd} of~\ref{procedure:bdf})
is appended into a single string.
The $q$-grams that are represented in the branching edges of $\Upsilon$
can be represented in the single string, by redundantly adding
$\mathit{suf}(X_{r(i)}([1:l]),q-1)$ in front of the string corresponding to
the next branch.
This leads to some duplicate partial decompression, but the resulting
string is still always shorter than the string produced by our
previous algorithm~\cite{goto11:_fast_minin_slp_compr_strin}.
The partial decompression of $X_{r(i)}([1:l])$ is implemented using
a simple $O(h+l)$ algorithm, where $h$ is the height of the SLP
which can be as large as $O(n)$.
All computations were conducted on a Mac Pro (Mid 2010)
with MacOS X Lion 10.7.2,
and 2 x 2.93GHz 6-Core Xeon processors and 64GB Memory,
only utilizing a single process/thread at once.
The program was compiled using the GNU C++ compiler ({\tt g++}) 4.6.2
with the {\tt -Ofast} option for optimization.
The running times were measured in seconds, after reading
the uncompressed text into memory for $\textrm{NSA}$, and
after reading the SLP that represents the text into memory for $\textrm{SSA}$ and
$\textrm{STSA}$.
Each computation was repeated at least 3 times, and the average was taken.
Table~\ref{table:running_time} summarizes the running times of the three algorithms.
$\textrm{SSA}$ and $\textrm{STSA}$ computed weighted $q$-gram frequencies on $t_i$ and $\Upsilon$, respectively.
Since the difference between the total length of $t_i$ and
the size of $\Upsilon$ becomes larger as $q$ increases,
$\textrm{STSA}$ outperforms $\textrm{SSA}$ when the value of $q$ is not small.
In fact, in Table~\ref{table:running_time} SSA2 was faster than $\textrm{SSA}$
for all values of $q > 3$.
$\textrm{STSA}$ was even faster than $\textrm{NSA}$ on the XML data whenever $q \leq 20$.
What is interesting is that $\textrm{STSA}$ outperformed $\textrm{NSA}$ on the ENGLISH data
when $q = 100$.
\begin{table}[t]
\caption{
Running time in seconds for SLPs that represent textual data from Pizza \& Chili Corpus.
The SLPs were constructed by RE-PAIR~\cite{LarssonDCC99}.
Bold numbers represent the fastest time for each data and $q$.
$\textrm{STSA}$ is faster than $\textrm{SSA}$ whenever $q>3$.
}
\label{table:running_time}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{1pt}
\renewcommand{ptm}{ptm}
\renewcommand{phv}{phv}
\renewcommand{pcr}{pcr}
\normalfont
\input{table_time.tex}
\end{center}
\end{table}
\end{document}
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\begin{document}
\def\@arabic\c@footnote{}
\footnotetext{ \texttt{File:~\jobname .tex,
printed: \number\year-0\number\month-\number\day,
\thehours.\ifnum\theminutes<10{0}\fi\theminutes}
} \makeatletter\def\@arabic\c@footnote{\@arabic\c@footnote}\makeatother
\maketitle
\begin{abstract}
Our aim in this paper, is to establish certain new integrals for the the $(p,q)-$Mathieu--power series. In particular, we investigate the Mellin-Barnes type integral representations
for a particular case of thus special function. Moreover, we introduce the notion of the $(p,q)-$Mittag-Leffler functions and we present a relationships between thus two functions. Some other applications are proved, in particular two Tur\'an type inequalities for the $(p,q)-$Mathieu series are proved.
\end{abstract}
\section{\bf Introduction}
\setcounter{equation}{0}
The following familiar infinite series
\begin{equation}
S(r)=\sum_{n=1}^\infty\frac{2n}{(n^2+r^2)^2},
\end{equation}
is called a Mathieu series. It was introduced and studied by \'Emile Leonard Mathieu in his
book \cite{18} devoted to the elasticity of solid bodies. Bounds for this series
are needed for the solution of boundary value problems for the biharmonic equations in a two--dimensional rectangular domain, see \cite[Eq. (54), p. 258]{13}.
Several interesting problems and solutions dealing with integral representations and
bounds for the following slight generalization of the Mathieu series with a fractional
power:
\begin{equation}\label{0t1}
S_\mu(r)=\sum_{n=1}^\infty\frac{2n}{(n^2+r^2)^{\mu+1}},\;(\mu>0,\;r>0),
\end{equation}
can be found in the recent works by Diananda \cite{D}, Tomovski and Tren\v{c}evski \cite{TT}, Srivatava et al. \cite{SKZ}. In \cite{SKZ}, the authors derived the following new Laplace type integral representation via Schlomilch series:
\begin{equation}
S_\mu(r)=\frac{\sqrt{\pi}}{2^{\mu-\frac{1}{2}}\Gamma(\mu+1)}\int_0^\infty e^{-rt} \mathcal{K}_\mu(t)dt, \;\mu>\frac{3}{2},
\end{equation}
where
$$\mathcal{K}_\mu(t)=t^{\mu+\frac{1}{2}}\sum_{k=1}^\infty\frac{J_{\mu+\frac{1}{2}}(kt)}{k^{\mu-\frac{1}{2}}}$$
with $J_\mu(z)$ is the Bessel function. Motivated essentially by the works of Cerone and Lenard \cite{C}, Srivastava and Tomovski in \cite{ZY} defined a family of generalized Mathieu series
\begin{equation}\label{;,}
S_\mu^{(\alpha,\beta)}(r; \textbf{a})=S_\mu^{(\alpha,\beta)}(r; \{a_k\}_{k=0}^\infty)=\sum_{k=1}^\infty\frac{2a_k^\beta}{(a_k^\alpha+r^2)^\mu},\;(\alpha,\beta,\mu,r>0),
\end{equation}
where it is tacitly assumed that the positive sequence
$$\textbf{a}=\{a_k\}=\{a_1,a_2,...\},\;\textrm{such \;that\;}\lim_{k\longrightarrow\infty}a_k=\infty,$$
is so chosen that the in?nite series in de?nition (\ref{;,}) converges, that is, that the following
auxiliary series
$$\sum_{k=1}^\infty \frac{1}{a_k^{\mu\alpha-\beta}},$$
is convergent.
\begin{define} $($see \cite[Eq. (6.1), p. 256]{SR}$)$ The extended beta function $B_{p,q}(x,y)$ is defined by
\begin{equation}\label{1}
B_{p,q}(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}E_{p,q}(t)dt,\;x,y,p,q\in\mathbb{C},\Re(p),\Re(q)>0,
\end{equation}
where $E_{p,q}(t)$ is defined by
$$E_{p,q}(t)=\exp\left(-\frac{p}{t}-\frac{q}{1-t}\right),\;p,q\in\mathbb{C},\Re(p),\Re(q)>0.$$
\end{define}
In particular, Chaudhry et al. \cite[p. 20, Eq. (1.7)]{CH}, introduced the $p–$extension of the Eulerian Beta function $B(x, y):$
$$B_p(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}e^{-\frac{p}{t(1-t)}}dt,\;\Re(p)>0,$$
whose special case when $p=0$ ( or $p=q=0$ in (\ref{1}) )we get the familiar beta integral
\begin{equation}
B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt,\;\Re(x),\Re(y)>0.
\end{equation}
\begin{define} $($see \cite[p. 4, Eq. 2.1]{23}$)$ Assume that $\lambda,\mu,s,p,q\in\mathbb{C}$ such that $\Re(p),\Re(q)\geq0$ and $\nu,a\in\mathbb{C}\setminus\mathbb{Z}_0^-.$ The extended Hurwitz-Lerch zeta function is defined by
\begin{equation}\label{2}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\sum_{n=0}^\infty\frac{(\lambda)_n}{n!}\frac{B_{p,q}(\mu+n,\nu-\mu)}{B(\mu,\nu-\mu)}\frac{z^n}{(a+n)^s},\:(|z|<1),
\end{equation}
where $(\lambda)_n$ denotes the Pochhammer symbol (or the shifted factorial) defined, in terms of Euler's Gamma function, by
\begin{displaymath}
(\lambda)_\mu=\frac{\Gamma(\lambda+\mu)}{\Gamma(\lambda)}=\left\{ \begin{array}{ll}
1& \textrm{$(\mu=0;\lambda\in\mathbb{C}\setminus\{0\})$}\\
\lambda(\lambda+1)...(\lambda+n-1)& \textrm{$(\mu=n\in\mathbb{N};\lambda\in\mathbb{C})$}
\end{array} \right.
\end{displaymath}
\end{define}
Upon setting $\lambda=1,$ (\ref{2}) reduces to
$$\Phi_{\mu,\nu}(z,s,a;p,q)=\sum_{n=0}^\infty\frac{B_{p,q}(\mu+n,\nu-\mu)}{B(\mu,\nu-\mu)}\frac{z^n}{(a+n)^s},\:(|z|<1).$$
It is easy to observe that
\begin{equation}\label{ZZ}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\frac{1}{\Gamma(\lambda)}D_z^{\lambda-1}\{z^{\lambda-1}\Phi_{\mu,\nu}(z,s,a;p,q)\},\;(\Re(\lambda)>0),
\end{equation}
where $D_z^{\lambda}$ denotes the well-known Riemann-Liouville fractional derivative operator defined by
\begin{equation}
D_z^{\lambda} f(z)=\left\{ \begin{array}{ll}
\frac{1}{\Gamma(-\lambda)}\int_0^z(z-t)^{-\lambda-1}f(t)dt& \textrm{$(\Re(\lambda)<0)$}\\
\frac{d^m}{dz^m}D_z^{\lambda-m} f(z)& \textrm{$(m-1\leq\Re(\lambda)<m,\;(m\in\mathbb{N}))$}
\end{array} \right.
\end{equation}
In \cite [Theorem 3.8]{LU1} Luo et al. proved the following integral representation for the extended Hurwitz-Lerch zeta funtion $\Phi_{\lambda,\mu,\nu}(z,s,a;p,q):$
\begin{equation}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1} e^{-at}{}_2F_1\Big[^{\;\lambda,\;\mu}_{\;\;\nu};ze^{-t};p,q\Big]dt,\;|z|<1,
\end{equation}
$$\left(p,q,a,s>0,\lambda,\mu\in\mathbb{C},\nu\in\mathbb{C}\setminus\mathbb{Z}_0^-\right),$$
where ${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p,q\Big]$ is the extended Gauss hypergeometric function defined by
$${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p,q\Big]=\sum_{n=0}^\infty (a)_n\frac{B_{p,q}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\;|z|<1,$$
$$\Big(\Re(p),\Re(q)\geq0, a, b\in\mathbb{C}, c\in\mathbb{C}\setminus\mathbb{Z}_0^-,\;\Re(c)>\Re(b)>0\Big).$$
When $p=q$ we obtain the extended of the extended of the Gaussian hypergeometric function $F_p$ defined by \cite{CH}:
$${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p\Big]=\sum_{n=0}^\infty (a)_n\frac{B_{p}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\;|z|<1,$$
$$\Big(\Re(p)\geq0, a, b\in\mathbb{C}, c\in\mathbb{C}\setminus\mathbb{Z}_0^-,\;\Re(c)>\Re(b)>0\Big).$$
The Fox-Wright function ${}^p\Psi_q[.]$ with $p$ numerator parameters $\alpha_1,...,\alpha_p$ and $q$ denominator parameters $\beta_1,...,\beta_q$ which are defined by
\begin{equation}\label{3}
{}_p\Psi_q\Big[_{(\beta_1,B_1),...,(\beta_q,B_q)}^{(\alpha_1,A_1),...,(\alpha_p,A_p)}\Big|z \Big]={}_p\Psi_q\Big[_{(\beta_q,B_q)}^{(\alpha_p,A_p)}\Big|z \Big]=\sum_{k=0}^\infty\frac{\prod_{l=1}^p\Gamma(\alpha_l+kA_l)}{\prod_{j=1}^q\Gamma(\beta_l+kB_l)}\frac{z^k}{k!},
\end{equation}
The defining series in (\ref{3}) converges in the whole complex $z-$plane when
$$\Delta=\sum_{j=1}^q B_j-\sum_{j=1}^p A_j>-1;$$
when $\Delta= 0$, then the series in (\ref{3}) converges for $|z|<\nabla,$ where
$$\nabla=\left(\prod_{j=1}^p A_j^{-A_j}\right)\left(\prod_{j=1}^qB_j^{B_j}\right).$$
If, in the definition (\ref{3}), we set
$$A_1=...=A_p=1\;\;\;\textrm{and}\;\;\;B_1=...=B_q=1,$$
we get the relatively more familiar generalized hypergeometric function ${}_pF_q[.]$ given by
\begin{equation}\label{hyper}
{}_p F_q\left[^{\alpha_1,...,\alpha_p}_{\beta_1,...,\beta_q}\Big|z\right]=\frac{\prod_{j=1}^q\Gamma(\beta_j)}{\prod_{i=1}^p\Gamma(\alpha_i)}{}_p\Psi_q\Big[_{(\beta_1,1),...,(\beta_q,1)}^{(\alpha_1,1),...,(\alpha_p,1)}\Big|z \Big]
\end{equation}
In this paper we consider the $(p,q)-$Mathieu type power series defined by:
\begin{equation}\label{*}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n B_{p,q}(\tau+n,\omega-\tau)z^n}{n!B(\tau,\omega-\tau)(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\Re(p),\Re(q)\geq0,\;|z|\leq1\right).$$
In particular case when $p=q,$ we define the $p-$Mathieu type power series defined by:
\begin{equation}\label{**}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n B_{p}(\tau+n,\omega-\tau)z^n}{n!B(\tau,\omega-\tau)(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\Re(p)\geq0,\;|z|\leq1\right).$$
The function $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ has many other special cases. We set $p=q=0$ we get
\begin{equation}\label{**}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};z)=S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};0,0;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n (\tau)_nz^n}{n!(\omega)_n(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu,\tau,\omega>0,\;|z|\leq1\right).$$
On the other hand, by letting $\tau=\omega$ in (\ref{**}) we obtain \cite[Eq. 5, p. 974]{ZK}:
\begin{equation}
S_{\mu,\nu}^{(\alpha,\beta)}(r;\textbf{a};z)=S_{\mu,\nu,\tau,\tau}^{(\alpha,\beta)}(r;\textbf{a};z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n z^n}{n!(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\;|z|\leq1\right).$$
Furthermore, the special cases when $\nu=z=1$ we get the generalized Mathieu series (\ref{;,}).
The contents of our paper is organized as follows. In section 2, we present new integral representation for the $(p,q)-$Mathieu series. In particular, we derive the Mellin-Barnes type integral representations
for $(p,q)-$Mathieu series $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).$ As applications, In Section 3, we introduce the $(p,q)-$Mittag-Leffler functions and we derive some relationships between thus two special functions, in particular we derive new series representations for the $(p,q)-$Mathieu series. Relationships between the $(p,q)-$ and generalized Mathieu series are proved and two Tur\'an type inequalities are established.
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\section{\bf Integral representation for the $(p,q)-$Mathieu types series}
In the course of our investigation, one of the main tools is the following result providing the integral representation for the $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z).$
\begin{theorem}\label{T1}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0$ such that $\gamma(\mu\alpha-\beta)>0.$ Then $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z)$ possesses the integral representation given by:
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z)=$$
\begin{equation}\label{a}
=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{theorem}
\begin{proof}By using the definition (\ref{*}), we can write the extended Mathieu types series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ in the following form:
\begin{equation}\label{mmm}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=1}^\infty\frac{(\nu)_n}{a_n^{(\mu+m)\alpha-\beta}}\frac{B_{p,q}(\tau+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{n!}.
\end{equation}
Therefore,
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{k^\gamma\}_{k=0}^\infty;p,q;z\Big)=$$
\begin{equation*}
\begin{split}
\;\;\;\;&=2z\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu)_{n+1}}{(n+1)!}\frac{B_{p,q}(\tau+1+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{(n+1)^{\gamma((\mu+m)\alpha-\beta})}\\
\;\;\;\;&=2\nu z\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu+1)_{n}}{n!}\frac{B_{p,q}(\tau+1+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{(n+1)^{\gamma((\mu+m)\alpha-\beta)+1}}\\
\;\;\;\;&=\frac{2\nu zB(\tau+1,\omega-\tau)}{B(\tau,\omega-\tau)} \sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu+1)_{n}B_{p,q}(\tau+1+n,\omega-\tau)z^n}{n!B(\tau+1,\omega-\tau)(n+1)^{\gamma((\mu+m)\alpha-\beta)+1}}\\
\;\;\;\;&=\frac{2\nu z B(\tau+1,\omega-\tau)}{B(\tau,\omega-\tau)} \sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \Phi_{\nu+1,\tau+1,\omega+1}(z,\gamma[(\mu+m)\alpha-\beta]+1,1;p,q)\\
\;\;\;\;\;\;\;\;\;\;\;\;&=\frac{2\nu \tau z}{\omega}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\left[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\right]\left(\sum_{m=0}^\infty\frac{\binom{\mu+m-1}{m}(-r^2t^{\gamma\alpha})^m }{\Gamma(\gamma[(\mu+m)\alpha-\beta]+1)}\right)dt\\
\;\;\;\;\;\;\;\;&=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{split}
\end{equation*}
This completes the proof of Theorem \ref{T1}.
\end{proof}
Now, in ths case $p=q$, Theorem \ref{T1} reduces to the following corollary.
\begin{coro}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;\Re(p)\geq0$ such that $\gamma(\mu\alpha-\beta)>0.$ Then $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p;z)$ possesses the integral representation given by:
\begin{equation}\label{a}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p;z)=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{coro}
\begin{remark}
1. By letting $p=q=0$ in (\ref{a}), we deduce that the function $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)$ possesses the following integral representation:
\begin{equation}\label{b}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t}\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
2. Setting $\tau=\omega$ in (\ref{b}) and using the fact that
$${}_2F_1\Big[^{\;a,\;b}_{\;\;b};z\Big]=(1-z)^{-a}$$
we obtain the following integral representation for the function $S_{\mu,\nu}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)$\cite[Theorem 1, Eq. 8]{ZK}
\begin{equation}
S_{\mu,\nu}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)=\frac{2\nu z}{\Gamma(\mu)}\int_0^\infty \frac{t^{\gamma[(\mu\alpha-\beta]}e^{-t}}{(1-ze^{-t})^{\nu+1}}{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{remark}
In the next Theorem we present the Mellin-Barnes integral representation for the alternating Mathieu-series $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).$
\begin{theorem}\label{TTT222222222222}Let $r,\nu,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0.$ Then the following integral representation
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big)=$$
\begin{equation}
=-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p,q}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds,
\end{equation}
holds true for all $|\arg(-z)|<\pi$.
\end{theorem}
\begin{proof}The contour of integration extends from $c-i\infty$ to $c+i\infty,$ such that all the poles of the Gamma function $\Gamma(\nu-s+1)$ at the points $s=k+\nu+1,\;k\in\mathbb{N}$ are separated from the poles of the gamma function $\Gamma(s)$ at the points $s=-k,\; k\in\mathbb{N}.$ Suppose that the the poles of the integrand are simple and using the fact that
$$ \textrm{res}[\Gamma,-k] =\lim_{s\longrightarrow -k}(s+k)\Gamma(s)=\frac{(-1)^k}{k!},$$
we find that
$$\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p,q}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds$$
\begin{equation*}
\begin{split}
&=\frac{2z}{\Gamma(\nu)}\sum_{k=0}^\infty \lim_{s\longrightarrow -k}\frac{(s+k)\Gamma(s)B_{p,q}(\tau-s+1,\omega-\tau)\Gamma(\nu-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}\\
&=\frac{2z}{\Gamma(\nu)}\sum_{k=0}^\infty \frac{(-1)^k}{k!}\frac{B_{p,q}(\tau+k+1,\omega-\tau)\Gamma(\nu+k+1)}{B(\tau,\omega-\tau)((k+1)^2+r^2)^\mu}z^k\\
&=-2\sum_{k=1}^\infty\frac{(\nu)_k k B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)(k^2+r^2)^\mu}\frac{(-z)^k}{k!}\\
&=-S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).
\end{split}
\end{equation*}
This completes the proof of Theorem \ref{TTT222222222222}
\end{proof}
\begin{coro}\label{cccc}
Let $r,\nu,\mu,\tau,\omega>0,\;\Re(p)\geq0.$ Then the following integral representation
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p;-z\Big)=$$
\begin{equation}
=-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds,
\end{equation}
holds true for all $|\arg(-z)|<\pi$.
\end{coro}
\begin{remark}If we set $p=0$ in Corollary \ref{cccc}, then we get the Mellin-Barnes representation of the function $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big):$
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big)=$$
\begin{equation}
=-\frac{z\Gamma(\omega)}{i\pi\Gamma(\nu)\Gamma(\tau)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)\Gamma(\tau-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{\Gamma(\omega-s+1)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds
\end{equation}
In particular, for $\tau=\omega$ we get
$$S_{\mu,\nu}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big)=
-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds.$$
Moreover, if we set $\mu=2$ and $\nu=1$ in the above equation we get the Mellin-Barnes for the alternating Mathieu-series proved by Saxena el al. \cite[Theorem 3.1]{SA}.
\end{remark}
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\section{\bf Applications}
In our first application in this section we present the relationships between the $(p,q)-$Mathieu-type series $S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)$ and the Rieman-Liouvile operator.
\subsection{Relationships with $(p,q)-$Mathieu-type series and the Rieman-Liouvile operator}
Our first main application is asserted by the following Theorem.
\begin{theorem} Let $r,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0$ and $0\leq\nu<1$. Then
\begin{equation}
S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)=\frac{1}{2ir\Gamma(\nu)}\left\{D_z^{\nu-1}\left(z^{\nu-1}\Phi_{\tau,\omega}(z,2,-ir;p,q)\right)-D_z^{\nu-1}\left(z^{\nu-1}\Phi_{\tau,\omega}(z,2,ir;p,q)\right)\right\}.
\end{equation}
\end{theorem}
\begin{proof}By using the definition of the $(p,q)-$Mathieu-type series, we can write the Mathieu-type series\\ $S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)$ in the following form:
\begin{equation}
S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)=\frac{1}{2ir}\left[\Phi_{\nu,\tau,\omega}(z,2,-ir;p,q)-\Phi_{\nu,\tau,\omega}(z,2,ir;p,q)\right].
\end{equation}
Combining the above equation with (\ref{ZZ}), we get the desired result.
\end{proof}
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\subsection{Relationships with $(p,q)-$Mittag-Leffler function and $(p,q)-$Mathieu-type series} In this section, we introduce the definition of the $(p,q)-$Mittag-Leffler function and we establish an integral representation for this function and we present some relationships with the $(p,q)-$Mathieu-type series.
For $\lambda,\tau,\omega,\theta,\sigma, \delta>0$ and $\Re(p),\Re(q)\geq0$ we define the $(p,q)-$Mittag-Leffler function by
\begin{equation}\label{def1}
E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}, z\in\mathbb{C}.
\end{equation}
In the case $p=q$ we define the $p-$Mittag-Leffler function by
\begin{equation}
E_{\delta,\theta,\sigma;p}^{(\lambda,\tau,\omega)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}, z\in\mathbb{C},
\end{equation}
whose special case when $p=0$ reduces to the generalized Mittag-Leffler function, introduced by Tomovski and Mehrez in \cite{ZK}
\begin{equation}
E_{\delta,\theta,\sigma}^{(\lambda)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{z^k}{k!}, z\in\mathbb{C}.
\end{equation}
For $\lambda=1$ the above series was introduced by S. Gerhold \cite{GG}.
\begin{lemma}\label{LLLL}For $\tau,\omega,\theta,\sigma,\delta>0$ and $\Re(p),\Re(q)\geq0.$ Then we have
\begin{equation}\label{55555}
E_{\delta,\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z)=\frac{\omega}{z\tau}\left[E_{\delta,\theta,\sigma;p,q}^{(1,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\delta}B(\tau,\omega-\tau)}\right].
\end{equation}
\end{lemma}
\begin{proof}By computation, we get
\begin{equation*}
\begin{split}
E_{\delta,\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z)&=\sum_{k=0}^\infty \frac{B_{p,q}(\tau+k+1,\omega-\tau)z^k}{B(\tau+1,\omega-\tau)\left[\Gamma(\theta k+\sigma+\theta)\right]^\delta}\\
&=\frac{B(\tau,\omega-\tau)}{zB(\tau+1,\omega-\tau)}\sum_{k=1}^\infty \frac{B_{p,q}(\tau+k,\omega-\tau)z^k}{B(\tau,\omega-\tau)\left[\Gamma(\theta k+\sigma)\right]^\delta}\\
&=\frac{\omega}{z\tau}\left[E_{\delta,\theta,\sigma;p,q}^{(1,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\delta}B(\tau,\omega-\tau)}\right].
\end{split}
\end{equation*}
The proof of Lemma \ref{LLLL} is completes.
\end{proof}
\begin{theorem}\label{8888}Let $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$Then the $(p,q)-$Mathieu-type series admits the following series representation:
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)$$
\begin{equation}\label{--}
\begin{split}
\qquad\qquad\qquad\qquad=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma;p,q}^{(\nu,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\gamma[(\mu+m)\alpha-\beta]}B(\tau,\omega-\tau)}\right].
\end{split}
\end{equation}
Moreover, the following series representation
\begin{equation}\label{;;}
S_{\mu,1 ,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=\frac{2z\tau}{\omega}\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z).
\end{equation}
holds true.
\end{theorem}
\begin{proof} In view of the definition of the $(p,q)-$Mittag-Leffler function (\ref{def1}) and the equation (\ref{mmm}) we obtain (\ref{--}). Finally, combining the equation (\ref{--}) with the following relation (\ref{55555}) we obtain the formula (\ref{;;}).
\end{proof}
Taking in (\ref{--}) the values $\theta=\sigma=1$ we obtain the following representation:
\begin{coro}Let $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$Then the $(p,q)-$Mathieu-type series admits the following series representations:
\begin{equation}
\begin{split}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{(k!)^\gamma\}_{k=0}^\infty;p,q;z\Big)=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[E_{\gamma[(\mu+m)\alpha-\beta],1,1;p,q}^{(\nu,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}\right],
\end{split}
\end{equation}
and
\begin{equation}
S_{\mu,1 ,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{k!^\gamma\}_{k=0}^\infty;p,q;z\Big)=\frac{2z\tau}{\omega}\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],1,2;p,q}^{(1,\tau+1,\omega+1)}(z).
\end{equation}
\end{coro}
\begin{lemma}\label{L1}For $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$ Then the the $(p,q)-$Mittag-Leffler function $E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)$ possesses the following integral representation:
\begin{equation}\label{int1}
E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)=\frac{1}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\delta,\theta,\sigma}^{(\lambda)}(zt)dt,
\end{equation}
holds true.
\end{lemma}
\begin{proof}By using the definition of the $(p,q)-$Beta function we get
\begin{equation*}
\begin{split}
\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\delta,\theta,\sigma}^{(\lambda)}(zt)dt&=\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\left(\sum_{k=0}^\infty\frac{(\lambda)_k(zt)^k}{\left[\Gamma(\theta k+\sigma)\right]^\delta k!}\right)dt\\
&=\sum_{k=0}^\infty\frac{(\lambda)_k z^k}{\left[\Gamma(\theta k+\sigma)\right]^\delta k!}\int_0^1 t^{\tau+k-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)dt\\
&=B(\tau,\omega-\tau)\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}\\
&=B(\tau,\omega-\tau)E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z).
\end{split}
\end{equation*}
The proof of Lemma \ref{L1} is completes.
\end{proof}
\begin{theorem}\label{T5}For $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$ Then the following integral representation
\begin{equation}\label{HHHHH}
\begin{split}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,\nu}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(\tau^2+[\Gamma(\sigma)]^\alpha)^\mu}
\end{split}
\end{equation}
holds true for all $|z|<1.$ Moreover, the following integral representation
\begin{equation}\label{HHHHHH}
\begin{split}
S_{\mu,1,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=
\end{split}
\end{equation}
$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{2z\tau}{\omega B(\tau+1,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,1}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt,$$
holds true for all $|z|<1.$
\end{theorem}
\begin{proof}By means of Lemma \ref{L1} and the integral representation (\ref{--}), we get
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=$$
\begin{equation*}
\begin{split}
&=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[\frac{1}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma}^{(\nu)}(zt)dt\right]\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\mu\alpha-\beta}B(\tau,\omega-\tau)}\sum_{m=0}^\infty\binom{\mu+m-1}{m} \left(\frac{-\tau^2}{[\Gamma(\sigma)]^\alpha}\right)^m\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\left(\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma}^{(\nu)}(zt)\right)dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\mu\alpha-\beta}B(\tau,\omega-\tau)}.\frac{1}{(1+\frac{r^2}{[\Gamma(\sigma)]^\alpha})^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k}{[\Gamma(\theta k+\sigma)]^{\gamma(\mu\alpha-\beta)}}\left(\sum_{m=0}^\infty\frac{\binom{\mu+m-1}{m}(-r^2)^m}{[\Gamma(\theta k+\sigma)]^{\gamma m\alpha}}\right)\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k}{[\Gamma(\theta k+\sigma)]^{\gamma(\mu\alpha-\beta)}}\left(1+\frac{r^2}{[\Gamma(\theta k+\sigma)]^{\gamma\alpha}}\right)^{-\mu}\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k[\Gamma(\theta k+\sigma)]^{\gamma\beta}}{\left([\Gamma(\theta k+\sigma)]^{\gamma\alpha}+r^2\right)^\mu}\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(\tau^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,\nu}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt\\&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu},
\end{split}
\end{equation*}
which evidently completes the proof of the representation (\ref{HHHHH}). Finally, combining (\ref{int1}) and (\ref{;;}) and repeating the same calculations as above we get (\ref{HHHHHH}). The proof of Theorem \ref{T5} is completes.
\end{proof}
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\subsection{Tur\'an type inequalities for the $(p,q)-$Mathieu-type series }
\begin{theorem}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;p,q\geq0.$ Then the following assertions are true:
1. The $(p,q)-$Mathieu-type series considered as a function in $p$ $( $or $q)$ is completely monotonic and log-convex on $(0,\infty).$ Furthermore, the following Tur\'an type inequality
\begin{equation}\label{TURAN}
\left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p+1,q;z)\right]^2\leq S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p+2,q;z)
\end{equation}
holds true for all $z\in (0,1).$\\
2. Assume that $r^2+\textbf{a}\geq1.$ Then the $(p,q)-$Mathieu-type series considered as a function in $\mu$ is completely monotonic and log-convex on $(0,\infty).$ Furthermore, the following Tur\'an type inequality
\begin{equation}\label{TURAN1}
\left[S_{\mu+1,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)\right]^2\leq S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)S_{\mu+2,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)
\end{equation}
holds true for all $z\in (0,1)$ such that $r^2+\textbf{a}\geq1.$
\end{theorem}
\begin{proof}1. In \cite[Corollary 2.7]{LU1}, the authors proved that the extended beta function $p(\;\textrm{or}\;q)\mapsto B_{p,q} (x, y)$ is completely monotonic function on $(0,\infty)$ and using the fact that sums of completely monotonic functions are completely monotonic too, we deduce that the $p (\;\textrm {or}\;q)\mapsto S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ is completely monotonic and log-convex on $(0,\infty),$ since every completely monotonic function is log-convex ( see \cite[p.167]{WI}. Thus, for all $p_1,p_2>0,$ and $t\in[0,1],$ we obtain
\begin{equation*}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};tp_1+(1-t)p_2,q;z)\leq\left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p_1,q;z)\right]^t \left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p_2,q;z)\right]^{1-t}.
\end{equation*}
Letting $t=\frac{1}{2},\;p_1=p$ and $p_2=p+2$ in the above inequality we get the Tur\'an type inequality (\ref{TURAN}).\\
2. We note that the function $\mu\mapsto (r^2+\textbf{a})^{-\mu}$ is completely monotonic on $(0,\infty)$ such that $r^2+\textbf{a}\geq1,$ and consequently the function $\mu\mapsto S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ is completely monotonic and log-convex on $(0,\infty).$
\end{proof}
\begin{remark}The condition $r^2+\textbf{a}$ is not necessary for proved the Tur\'an type inequality (\ref{TURAN1}), a similar proof of the Theorem 2 in \cite{SKZ}, we obtain (\ref{TURAN1}).
\end{remark}
\end{document}
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\begin{document}
\title{Filtering problems with exponential criteria for general Gaussian signals}
\begin{abstract} The explicit solution of the discrete time filtering problems with exponential criteria for a general
Gaussian signal is obtained through an approach based on a conditional Cameron-Martin type formula.
This key formula
is derived for conditional expectations
of exponentials of some quadratic forms of Gaussian sequences. The formula
involves conditional expectations
and conditional covariances in some auxiliary optimal risk-neutral filtering problem
which is used in the proof. Closed form recursions of Volterra
type for these ingredients are provided. Particular cases for which the results can be further
elaborated are investigated.
\end{abstract}
\vspace*{4mm}
\paragraph{Key words.}{Gaussian process, optimal
filtering, filtering error,
Riccati-Volterra equation, risk-sensitive filtering, exponential criteria }
\vspace*{0.25cm}
\paragraph{AMS subject classifications.} Primary 60G15. Secondary 60G44, 62M20.
\section{Introduction}
The linear exponential Gaussian (LEG for short) filtering problem,
\textit{i.e.}, with an exponential cost criteria (see the definition
(\ref{LEGdef}) below), and the so called risk-sensitive (RS for short) filtering
problem (see \cite{collings} and the statement (\ref{rde})
below) have been given a great deal of interest over the last
decades. Numerous results have been already reported in specific
models, specially around Markov models, but, as far as we know,
without exhibiting the relationship between these two problems.
See, \textit{e.g.}, Whittle \cite{whittle4}-\cite{whittle3}, Speyer \textit{et
al.} \cite{speyer}, Elliott \textit{et al}. \cite{elliott4}, \cite{dey1},
\cite{elliott1} and Bensoussan and van Schuppen
\cite{bensoussan} for contributions on this subject and related LEG and RS control problems. Therein
the notion of ``information state" has been introduced
without any clear
probabilistic meaning for auxiliary processes which are involved,
even in the Gauss-Markov case.
Moreover, the method proposed in \cite{elliott4} does not work in
a non Markovian situation.
In our paper \cite {AMM}, we have solved the LEG and RS filtering problems for general
Gaussian signal processes in continuous time and in the particular setting where the functional in the exponential is a \textit{singular} quadratic functional. Moreover we have proved that actually in this case the solutions coincide.
In our paper \cite{AMM2} we have solved the LEG and RS filtering problems for Gauss-Markov processes but with a \textit{nonsingular} quadratic functional in the exponential.
In this setting we have proposed an example to show that the solutions may be different.
On the other hand,
the general solution for the
optimal risk-neutral linear filtering problem and
a Cameron-Martin type formula for general Gaussian sequences have been obtained in \cite{mkalbmcr3}. It seems natural
to use the approach proposed in \cite{mkalbmcr3} and \cite{AMM} to derive the solution of the
LEG and RS filtering problems for general Gaussian signals in discrete time setting, to precise their link and also to give a
probabilistic interpretation for the ingredients of the ``information state".
In the present paper we are interested in the explicit solution of the Linear Exponential Gaussian (LEG) and Risk Sensitive (RS) filtering problems for general Gaussian signals. Namely we deal with a signal-observation model
$
(X_t,Y_t)_{t\ge 1},
$
where the signal $X=(X_t)_{t\ge 1}$ is an arbitrary Gaussian sequence with mean
$m=(m_t, t\geq 1)$ and covariance $K =(K(t,s), t\geq 1,
s\geq 1)$, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg X_t=m_t,\quad\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)(X_s-m_s)=K(t,s)\,, \quad
t\geq 1\,,\; s\geq 1\,,
$$
and, for some sequence $A=(A_t,\, t\ge 1)$ of the real numbers, the observation process $Y=(Y_t,\, t\ge 1)$ is given by
\begin{equation}\label{observ}
Y_t= A_t X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t,
\end{equation}
where $\mathop{\mathrm{var}}\nolimitsepsilon=(\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d. $\mathcal{N}(0,1)$ random variables and $\mathop{\mathrm{var}}\nolimitsepsilon$ and $X$ are independent.
Suppose that only $Y$ is observed and for a given real number $\mu$ and a fixed sequence $(Q_{t})_{t\ge 1}$ of nonnegative real numbers, one wishes
to minimize with respect to $h:\,h_t\in {\cal Y}_{t}, t\ge 1$ the quantity:
\begin{equation}\label{LEGcrit}
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\},
\end{equation}
where $({{\cal Y}}_t)$ is the natural filtration of $Y$, \textit{i.e.},
${{\cal Y}}_t=\sigma(\{Y_u\, ,\, 1\leq u\leq t\})$ and $h_t\in{{\cal Y}}_t$ means that $h_t$ is ${{\cal Y}}_t$-measurable.
Note
that, according to the sign of the real parameter $\mu$, there are two different
cases for this linear exponential Gaussian (LEG)
filtering problem (the terminology is taken from the linear exponential Gaussian optimal control problem) :
\begin{itemize}
\item$\mu < 0 ,$ called \textit{risk-preferring} filtering problem,
\item $\mu > 0, $ called the \textit{risk-averse} filtering problem.
\end{itemize}
It is well known (see, \textit{e.g.}, \cite{speyer} for the
Markov case) that the solution to this problem is
not the conditional expectation of $X_t$ given the $\sigma$-field
${\cal Y}_t$. Our first aim is to show that the solution
can be completely explicited~: the characteristics of the optimal
solution are obtained as the solution of a closed form system of
Volterra type equations which actually reduce to the
equations known also for the RS setting when the signal process $X$ is
Gauss-Markov (see, \textit{e.g.}, \cite{elliott2}). Our second aim is to give
the probabilistic interpretation of this optimal solution in terms of an
auxiliary \textit{risk-neutral } filtering problem. Actually, we extend the
filtering approach initiated in \cite{mkalbmcr3} and \cite{AMM} for
one-dimensional processes, to obtain a conditional Cameron-Martin
type formula for the {\em conditional Laplace transform} of a
quadratic functional of the involved process. Namely, we give an
explicit representation for the random variable
\begin{equation}\label{def:It}
\mathcal{I}_T= \mathop{\mathbb{E}}\nolimits \left(\left.\exp\left\{\frac{\mu}{2} \sum_{s=1}^T (X_s-h_s)^2 Q_s \right\} \right/ {{\cal Y}}_{T}\right),
\end{equation}
where $h_s\in {\cal Y}_{s}, \, s\ge 1$.
The paper is organized as follows. In Section \ref{LEG} we derive
the solution of the LEG filtering problem : explicit recursive
equations, involving the covariance function of
the filtered process, are obtained. In particular, in Section \ref{CMC}, an appropriate auxiliary risk-neutral filtering problem is
matched to that of deriving the key
Cameron-Martin type formula. The solution of this auxiliary filtering problem is discussed
in Section \ref{AFP}. In Section \ref{PC} we investigate some specific cases
where the results can be further elaborated. In Section \ref{disc} we discuss the relationship between LEG and
RS filtering problems. Section \ref{interpret}
is devoted to the interpretation for the ingredients of the ``information state''.
Finally, Sections \ref{complements} and \ref{againpart} are devoted to a more general case, namely when the particular structure of the observation sequences $(Y_t)_{t\ge 1}$ is not specified.
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\section{Solution of the LEG filtering problem}\label{LEG}
Let us introduce the following condition $(C_{\mu})$:
\begin{enumerate}
\item[$(C_{\mu})$]
the equation
\begin{equation}\label{GAMMABAR}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} \overline{\gamma}(t,l)\overline{\gamma}(s,l) \, \frac{S_l}{1+S_l \overline{\gamma}_l},\quad S_l=A_l^2-\mu Q_l
\end{equation}
has a unique and bounded solution on $\{(t,s):1\le s \le t \le T\}$, such that $\overline{\gamma}_l=\overline{\gamma}(l,l) \ge 0,\, l\ge 1$ and moreover
$$ \displaystyle{1 +
S_{l}\overline{\gamma}_l}> 0,\, l\ge 1.
$$
\end{enumerate}
\begin{rem}
Notice that for all $\mu$ \textbf{negative} the condition $(C_{\mu})$ is satisfied and if $\mu $ is \textbf{positive}, the condition $(C_{\mu})$ is satisfied for $\mu$ sufficiently small, for example, those such that for any
$ t \le T \, A_{t}^{2}-\mu Q_{t}$ is nonnegative.
\end{rem}
The first result is the following
\begin{theorem}\label{LEGsol}
Suppose that the condition $(C_{\mu})$ is satisfied. Let $(\overline{h}_t)_{t\ge 1}$ be the solution of the following equation:
\begin{equation}\label{hbar}
\overline{h}_t=m_t + \sum_{l=1}^t A_l\overline{\gamma}(t,l) (Y_l-A_l \overline{h}_l),
\end{equation}
where $\overline{\gamma}=(\overline{\gamma}(t,s), 1\le s \le t \le T)$ is the unique solution of equation \eqref{GAMMABAR}.\\
Then $(\overline{h}_t)_{t\ge 1}$ is the solution of the LEG filtering problem,
\textit{i.e.},
\begin{equation}\label{LEGdef}
\overline{h}=\mathop{\mathrm{argmin}}_{h:\,h_t\in {\cal Y}_{t}, t\ge 1} \mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}.
\end{equation}
Moreover, the corresponding optimal risk is given by
$$
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-\overline{h}_t)^2 Q_t \right\}
=\mu \prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2}.
$$
\end{theorem}
Theorem \ref{LEGsol} is a direct consequence of
results of Section \ref{CMC}. Its proof will be given at the end of
Section \ref{CMC}.
\begin{rem}
\begin{itemize}
\item Note that equation \eqref{hbar} is really recursive equation and it can be rewritten in the equivalent form:
$$
\overline{h}_t=\frac{1}{1+A_t^2\overline{\gamma}_{t}}\left[m_t + \sum_{l=1}^{t-1} A_l\overline{\gamma}(t,l) (Y_l-A_l \overline{h}_l)+ A_t \overline{\gamma}_{t}Y_{t}\right],
$$
\item It is worth emphasizing that taking $\mu=0$ in equation
(\ref{GAMMABAR}), one gets through equation (\ref{hbar}) the solution $\bar{h}$ of the
\textit{risk-neutral} filtering problem of the signal $X$ given
the observation $ Y$, \textit{i.e.}, $\bar{h}_{t}= \mathop{\mathbb{E}}\nolimitsg ( X_{t} / {\cal{ Y }}_{t})$ (see, \textit{e.g.}, \cite{mkalbmcr3}).
\end{itemize}
\end{rem}
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\subsection{Conditional version of a Cameron-Martin formula}\label{CMC}
The proof of Theorem~\ref{LEGsol} is based on the conditional version of the Cameron--Martin formula which provides
the conditional expectation $\mathcal{I}_t$ defined by \eqref{def:It}. Let
\begin{equation}\label{def:Jt}
J_t=\exp\left\{ -\frac{1}{2}\sum\limits_{s=1}^t (X_s- h_s)^2 Q_s \right\}.
\end{equation}
Then $\mathcal{I}_t=\pi_t (J_t)$,
where for any random variable $\eta$ such that
$\mathop{\mathbb{E}}\nolimitsg |\eta|<+\infty$, the notation
$\pi_t(\eta)$ is used for the conditional expectation of $\eta$ given the
$\sigma$-field
${\cal
Y}_t=\sigma(\{Y_s\, ,\, 1\leq s\leq t\}),$
$$
\pi_t(\eta) = \mathop{\mathbb{E}}\nolimitsg(\eta/{\cal Y}_t)\,.
$$
\begin{proposition}\label{p:CM}
Suppose that the condition $(C_{\mu})$ is satisfied. Let $(\overline{\gamma}(t,s),\, 1\le s \le t \le T)$ be the solution of equation \eqref{GAMMABAR} and $(Z_t^h,\,t\ge 1)$ be the solution of the following equation
\begin{equation}\label{eqzh}
Z_t^h = m_t -\sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{\mu Q_l}{1+S_l \overline{\gamma}_l} (h_l - Z_{l}^h) + \sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{A_l}{1+S_l \overline{\gamma}_l} (Y_l - A_l Z_{l}^{h}).
\end{equation}
Then the following representation of the random variable $\mathcal{I}_T$ defined by \eqref{def:It} holds for any $T\ge 1$:
$$
\mathcal{I}_T=\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_t - \frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t } \right]^{2}\right\} \times \mathcal{M}_T,
$$
where $(\mathcal{M}_T)_{T\ge 1}$ is a martingale defined by :
\begin{multline}\label{martin}
\mathcal{M}_T = \prod_{t=1}^T \left[\frac{(1+A_t^2 \gamma_t)}{1+A_t^{2} \overline{\gamma}_t}\right]^{1/2} \exp\left\{ \frac{A_t}{1+A_t^2 \overline{\gamma}_t} \, (Z_{t}^h - \pi_{t-1}(X_t) ) \nu_t - \right.
\\
\left. -\frac{1}{2} \cdot \frac{A_t^2}{1+A_t^2 \overline{\gamma}_t} \, (Z_{t}^h - \pi_{t-1}(X_t))^2
- \frac{1}{2} \cdot \frac{A_t^2 (\gamma_t - \overline{\gamma}_t)\cdot \nu_t^2}{(1+A_t^2 \overline{\gamma}_t) (1+A_t^2 \gamma_t)}
\right\},
\end{multline}
in terms of the innovation sequence $(\nu_t)_{t \ge 1}$:
$$
\nu_t=Y_t - A_{t}\pi_{t-1}(X_t);\quad \pi_{t-1}(X_t)=\mathop{\mathbb{E}}\nolimits (X_t / {\cal Y}_{t-1}),
$$
and of the variances of one-step prediction errors $(\gamma_t)_{t \ge 1}$:
$$
\quad \gamma_t = \mathop{\mathbb{E}}\nolimits (X_t-\pi_{t-1}(X_t))^2.
$$
\end{proposition}
\begin{rem}
\begin{enumerate}
\item
The probabilistic interpretation of the auxiliary processes $(Z_t^h)$ and $(\overline{\gamma}_t)_{t\ge 1}$ appearing in the Proposition~\ref{p:CM} will be clarified below.
\item Proposition~\ref{p:CM} reduces to the ordinary Cameron-Martin type formula (\textit{cf.} Theorem~1 \cite{mkalbmcr3}) for $h\equiv 0$ when $A_t=0,\,l\ge 1$ and hence $X$ and $Y$ are independent.
\end{enumerate}
\end{rem}
\paragraph{Proof of Proposition~\ref{p:CM}}
We will prove Proposition~\ref{p:CM} for $\mu<0$, namely $\mu=-1$. Then we can replace $Q$ by $-\mu Q$ and the statement of Proposition~\ref{p:CM} is still valid because of the analytical properties of the involved functions.
The proof of Proposition~\ref{p:CM} for $\mu=-1$ will be separated into two steps.
\textbf{I.} (Actually it is the discrete time analog for the general filtering theorem.) Since $h_t\in {\cal Y}_{t},\, t\ge 1,$ in the proof we can suppose that $h$ is a deterministic function.
First of all, we claim that for $J_t$, defined by \eqref{def:Jt}
\begin{equation}\label{pitJ}
\pi_t (J_t) = \left. \frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(\beta_t^y)} \right|_{y=Y_t},
\end{equation}
where $\beta_t^y = \exp (A_t X_t y - \frac{1}{2} A_t X_t^2)$.
Indeed, let us introduce the new probability measure $\hat{\Pg}$, defined by
$$
\frac{d\hat{\Pg}}{d\Pg} = \exp (-A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2).
$$
The classical Bayes formula gives that
$$
\pi_t(J_t) = \frac{\hat{\pi}_t (J_t \exp (A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t + \frac{1}{2} A_t^2 X_t^2))}{\hat{\pi}_t( \exp (A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t + \frac{1}{2} A_t^2 X_t^2))} =
\frac{\hat{\pi}_t (J_t \exp (A_t X_t Y_t - \frac{1}{2} A_t^2 X_t^2))}{\hat{\pi}_t (\exp (A_t X_t Y_t - \frac{1}{2} A_t^2 X_t^2))},
$$
where $\hat{\pi}_t(\cdot)$ denotes a conditional expectation with respect to ${\cal Y}_{t}$ under $\hat{\Pg}$.
Note that under $\hat{\Pg}$ the distribution of $(X_s, Y_r)_{s\le t, \, r\le t-1}$ is the same as under $\Pg$ and
$Y_t$ is a ${\cal N}(0,1)$ random variable independent of $(X_s, Y_r)_{s\le t, \, r\le t-1}$ .
To understand this point it is sufficient to write the following equality for the mutual characteristic function with arbitrary real numbers $(\alpha_{j}, \lambda_{j})$:
\begin{multline*}
\hat{\mathop{\mathbb{E}}\nolimits} \exp \left\{i\sum_{j=1}^t \alpha_j X_j + i\sum_{j=1}^t \lambda_j Y_j \right\} =
\\
=\mathop{\mathbb{E}}\nolimits \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} Y_t - A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2 \right\} = \\
= \mathop{\mathbb{E}}\nolimits \left( \mathop{\mathbb{E}}\nolimits \left. \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} Y_t - A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2 \right\} \right/ {\cal X }_{t}\right) =
\\
=\mathop{\mathbb{E}}\nolimits \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} A_t X_t - \frac{1}{2} A_t^2 X_t^2 + \frac{1}{2} (i\lambda_{t} - A_t X_t)^2 \right\} =
\\
= e^{-\frac{1}{2}\lambda_{t}^2} \mathop{\mathbb{E}}\nolimits\exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j \right\},
\end{multline*}
where ${\cal
X}_t$ is the $\sigma$-field
${\cal
X}_t=\sigma(\{X_s\, ,\, 1\leq s\leq t\})$.
Hence,
\begin{multline*}
\hat{\pi}_t (J_t \exp(A_t X_t Y_t - \frac{1}{2}A_t^2 X_t^2 )) =
\\
=\pi_{t-1} (J_t \exp (A_t X_t y - \frac{1}{2} A_t^2 X_t^2 ))|_{y=Y_t} =
\\
= \pi_{t-1} (J_t \beta_t^y)|_{y=Y_t}.
\end{multline*}
Similarly,
$$
\hat{\pi}_t \left(\exp (A_t X_t y - \frac{1}{2} A_t^2 X_t^2 )\right) = \left.\pi_{t-1} (\beta_t^y)\right|_{y=Y_t}\,,
$$
and hence \eqref{pitJ} holds.
\textbf{II.} In the second step we will calculate the ratio $\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}}$ which, due to \eqref{pitJ} can be rewritten as
\begin{equation}\label{ratio}
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})}=\left. \frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\beta_t^y)} \right|_{y=Y_t}.
\end{equation}
For this aim similarly to what we proposed in~\cite{mkalbmcr3} and \cite{AMM} we introduce the auxiliary processes $(Y_t^2)_{t\ge 1}$ and $(\xi_t)_{t\ge 1}$.
Let $\bar{\mathop{\mathrm{var}}\nolimitsepsilon} =(\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t\ge 1}$ be a sequence of i.i.d. $\mathcal{N}(0,1)$ random variables independent of $X$ and define $(Y_t^2, \xi_t)_{t\ge 1}$ by:
\begin{equation}\label{Yaux}
Y_t^2= Q_t(X_t-h_t) + \sqrt{Q_t}\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t,
\end{equation}
\begin{equation}\label{xi:eq}
\xi_t=\sum_{s=1}^{t}(X_s-h_s) Y_s^2.
\end{equation}
Now the following equality holds:
$$
\left.\frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})}\right|_{y=Y_t} = \left.\frac{\overline{\pi}_{t-1}(\exp \{-\frac{1}{2}Q_t (X_t-h_t)^2 - \xi_{t-1}\} \beta_t^y)}{\overline{\pi}_{t-1}(\exp(-\xi_{t-1}))}\right|_{y=Y_t},
$$
where $\overline{\pi}_t(\cdot)$ stands for a conditional expectation w.r.t. to the $\sigma$-field $\bar{\cal Y}_t=\sigma(\{Y_s, Y_s^2, {s\le t}\})$ under the initial measure $\Pg$.
Again the proof of this equality is based on the Bayes formula. Namely, let $\tilde{\Pg}$ be the new probability measure defined by
\begin{equation}\label{ptilde}
\frac{d\tilde{\Pg}}{d\Pg} = \rho_{t-1}=\exp\left\{ -\frac{1}{2} \sum_1^{t-1} Q_s (X_s-h_s)^2 -\sum_1^{t-1} \sqrt{Q_s} (X_s-h_s) \bar{\mathop{\mathrm{var}}\nolimitsepsilon}_s \right\}.
\end{equation}
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Similarly,
$$
\hat{\pi}_t \left(\exp (A_t X_t y - \frac{1}{2} A_t^2 X_t^2 )\right) = \left.\pi_{t-1} (\beta_t^y)\right|_{y=Y_t}\,,
$$
and hence \eqref{pitJ} holds.
\textbf{II.} In the second step we will calculate the ratio $\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}}$ which, due to \eqref{pitJ} can be rewritten as
\begin{equation}\label{ratio}
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})}=\left. \frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\beta_t^y)} \right|_{y=Y_t}.
\end{equation}
For this aim similarly to what we proposed in~\cite{mkalbmcr3} and \cite{AMM} we introduce the auxiliary processes $(Y_t^2)_{t\ge 1}$ and $(\xi_t)_{t\ge 1}$.
Let $\bar{\mathop{\mathrm{var}}\nolimitsepsilon} =(\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t\ge 1}$ be a sequence of i.i.d. $\mathcal{N}(0,1)$ random variables independent of $X$ and define $(Y_t^2, \xi_t)_{t\ge 1}$ by:
\begin{equation}\label{Yaux}
Y_t^2= Q_t(X_t-h_t) + \sqrt{Q_t}\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t,
\end{equation}
\begin{equation}\label{xi:eq}
\xi_t=\sum_{s=1}^{t}(X_s-h_s) Y_s^2.
\end{equation}
Now the following equality holds:
$$
\left.\frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})}\right|_{y=Y_t} = \left.\frac{\overline{\pi}_{t-1}(\exp \{-\frac{1}{2}Q_t (X_t-h_t)^2 - \xi_{t-1}\} \beta_t^y)}{\overline{\pi}_{t-1}(\exp(-\xi_{t-1}))}\right|_{y=Y_t},
$$
where $\overline{\pi}_t(\cdot)$ stands for a conditional expectation w.r.t. to the $\sigma$-field $\bar{\cal Y}_t=\sigma(\{Y_s, Y_s^2, {s\le t}\})$ under the initial measure $\Pg$.
Again the proof of this equality is based on the Bayes formula. Namely, let $\tilde{\Pg}$ be the new probability measure defined by
\begin{equation}\label{ptilde}
\frac{d\tilde{\Pg}}{d\Pg} = \rho_{t-1}=\exp\left\{ -\frac{1}{2} \sum_1^{t-1} Q_s (X_s-h_s)^2 -\sum_1^{t-1} \sqrt{Q_s} (X_s-h_s) \bar{\mathop{\mathrm{var}}\nolimitsepsilon}_s \right\}.
\end{equation}
Then $J_t \rho_{t-1}= \exp \{-\xi_{t-1} -\frac{1}{2} Q_t (X_t-h_t)^2 \}$ and $J_{t-1} \rho_{t-1}= \exp \{-\xi_{t-1}\}$. Thus
\begin{multline*}
\left.\frac{\overline{\pi}_{t-1} (\exp(-\xi_t - \frac{1}{2}Q_t (X_t-h_t)^2 ) \beta_t^y)}{\overline{\pi}_{t-1} (\exp(-\xi_{t-1}))}\right|_{y=Y_t}=
\\
= \left.\frac{\mathop{\mathbb{E}}\nolimits (J_t \beta_t^y \rho_{t-1}/\bar{{\cal Y}}_{t-1})}{\mathop{\mathbb{E}}\nolimits (\rho_{t-1}/\bar{{\cal Y}}_{t-1})}\cdot
\frac{\mathop{\mathbb{E}}\nolimits (\rho_{t-1}/\bar{{\cal Y}}_{t-1})}{\mathop{\mathbb{E}}\nolimits \exp(J_{t-1}\rho_{t-1})/\bar{{\cal Y}}_{t-1})}\right|_{y=Y_t}=
\\
= \left.\frac{\tilde{\mathop{\mathbb{E}}\nolimits}(J_t \beta_t^y/\bar{{\cal Y}}_{t-1})}{\tilde{\mathop{\mathbb{E}}\nolimits}(J_{t-1}/\bar{{\cal Y}}_{t-1})}\right|_{y=Y_t} =\left. \frac{\pi_{t-1} (J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})}\right|_{y=Y_t},
\end{multline*}
where the last equality holds because under the probability measure $\tilde\Pg$ the distribution of $(X_s, Y_s)_{s\le t}$
is the same as under the initial measure $\Pg$ and
$(X_s, Y_s)_{s\le t-1}$ is independent of $(Y_{s}^{2})_{s\leq t-1}$.
Finally we have proved the following:
\begin{equation}\label{derivlogar}
\frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})} = \left.\frac{\overline{\pi}_{t-1}(\exp\left[-\xi_{t-1}+A_t X_t y -\frac{1}{2} Q_t (X_t -h_t)^2 - \frac{1}{2}A_t^2 X_t^2\right])}{\overline{\pi}_{t-1}(\exp(-\xi_{t-1})) \pi_{t-1}(\beta_{t}^{y})}\right|_{y=Y_t}.
\end{equation}
At this point we will use the conditionally Gaussian properties of $(X_{t},\xi_{t-1})$ w.r.t. $\bar{{\cal Y}}_{t-1}$ and Lemma 11.6 \cite{lipshi1} which says that for a Gaussian pair $(U,V)$ with mean values $m_{_{U}},m_{_{V}}$, variances $\gamma_{_{U}},\gamma_{_{V}}$ and covariance $\gamma_{_{UV}}$
\begin{multline}\label{feqg}
\mathop{\mathbb{E}}\nolimits\exp\left\{-\frac{1}{2} DU^2 + \lambda_1 U - \lambda_2 V \right\} = (1+D\gamma_{_{U}})^{-1/2} \times
\\
\times \exp\left\{ -\lambda_2 m_{V} + \frac{\lambda_2^2}{2} \gamma_{_{V}} - \frac{1}{2} \cdot\frac{D}{1+D\gamma_{_{U}}} (m_{_{U}}-\lambda_2 \gamma_{_{UV}})^2 \right. +
\\
+\left. \frac{\lambda_1^2 \gamma_{_{U}} + 2\lambda_1 (m_{_{U}}-\lambda_2\gamma_{_{UV}})}{2(1+D\gamma_{_{U}})} \right\},
\end{multline}
for any real numbers $\lambda_{1},\lambda_{2}$ and $D\ge 0$.
Indeed, in \eqref{derivlogar} we will apply this formula to $(U,V)=(X_{t},\xi _{t-1})$ given $\bar{{\cal Y}} _{t-1}$ with
$$
D=S_{t}=Q_t+A_t^2, \quad \lambda_2=1, \quad \lambda_1=A_t y+ Q_t h_t,
$$
in the numerator and
$D=\lambda_1=0 ,\quad \lambda_2=1 $ in the first factor of the denominator and again to $(U,V)=(X_{t},\xi _{t-1})$ given ${{\cal Y}} _{t-1}$ with
$$
D=A_t^2, \quad \lambda_2=0, \quad \lambda_1=A_t y,
$$
in the second factor of the denominator.
Collecting the terms as coefficients for $h^{2}_{t}$ and $h_{t}$, we obtain that
\begin{multline*}
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{(1+S_{t}\overline{\gamma}_t)^{-1/2}}{(1+A_{t}^{2}\gamma_t)^{-1/2}} \cdot \exp \left\{-\frac{Q_{t}}{2} \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_{t}-\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+A_t^2 \overline{\gamma}_t} \right]^2 \right\} \times \\
\times \exp\left\{
-\frac{A_t^2 (Z_{t}^h)^2 - A_{t}^{2} \overline{\gamma}_{t} Y_t^2}{2(1+A_t^2 \overline{\gamma}_{t})} + \frac{Y_tZ_{t}^hA_t}{1+A_{t}^2 \overline{\gamma}_t} + \frac{1}{2} \cdot \frac{A_t^2 \pi_{t-1}^2(X_{t}) - 2 A_t \pi_{t-1}(X_{t}) Y_t - A_{t}^2 Y_t^2 \gamma_t}{1+A_{t}^2\gamma_t}
\right\},
\end{multline*}
where
$Z_{t}^h=\overline{\pi}_{t-1}(X_{t})-\overline{\gamma}_{_{X\xi}}(t)$ with
\begin{multline}\label{def gamma Xxi}
\overline{\gamma}_{_{X\xi}}(t)=\mathop{\mathbb{E}}\nolimitsg[(X_t-\overline{\pi}_{t-1}(X_t))(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t-1}],\,t\ge 2\,;\,
\overline{\gamma}_{_{X\xi}}(1)=0\,.
\end{multline}
To finish the proof we just replace $Y_t$ by $\nu_t+A_t \pi_{t-1}(X_{t})$. Thus in the last exponential term we find:
\begin{multline*}
\exp\left\{ - \frac{\nu_t^2 A_t^2 (\gamma_t-\overline{\gamma}_t)} {2(1+A_t^2 \overline{\gamma}_t)(1+A_t^2 \gamma_t)} + \frac{Z_{t}^{h}-\pi_{t-1}(X_{t})}{1+A_{t}^2\overline{\gamma}_t}A_t \nu_t \right. -
\\ - \left.\frac{1}{2}\cdot \frac{A_t^2}{1+A_t^2 \overline{\gamma}_t} (Z_{t}^h-\pi_{t-1}(X_{t}))^2 \right\},
\end{multline*}
which gives the Proposition.
\begin{rem}\label{probinterp}
\begin{enumerate}
\item
Note that now the probabilistic interpretation of the ingredients $\overline{\gamma}_t$ and $Z_t^{h}$ is clarified for \textbf{negative} $\mu$. Namely,
$\overline{\gamma}_t=\mathop{\mathbb{E}}\nolimits (X_t-\overline{\pi}_{t-1}(X_t))^2,$
and $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$, but when
$\mu$ is \textbf{positive}, there is no such connection anymore.
\item Observe that actually $\overline{\pi}_{t-1}(X_t)$ and $\overline{\gamma}_{_{X\xi}}(t)$ are $\bar{{\cal Y}}_{t-1}$-measurable, but the difference $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ is ${\cal Y}_{t-1}$ measurable.
\end{enumerate}
\end{rem}
\paragraph{Proof of Theorem \ref{LEGsol}} The statement of Theorem
\ref{LEGsol} is the direct consequence of
Proposition \ref{CMC}. Indeed, we claim that the following chain of inequalities holds for any $h:\,h_t\in {\cal Y}_{t}, t\ge 1$ :
$$
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
$$
$$
=\mathop{\mathbb{E}}\nolimits\left[\mathop{\mathbb{E}}\nolimits\mu \left(\left.\exp\left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\} \right/ {{\cal Y}}_{T}\right)\right]
$$
$$
=\mu\mathop{\mathbb{E}}\nolimits\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_t - \frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t } \right]^{2}\right\} \times \mathcal{M}_T,
$$
$$
\stackrel{(a)}{\ge}\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \mu \mathop{\mathbb{E}}\nolimits \mathcal{M}_T
$$
$$
\stackrel{(b)}{=} \mu \prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} .
$$
Of course under condition $(C_{\mu})$, since the term in the last line is finite, it is sufficient to consider the case:
\begin{equation}\label{expfin}
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
< \infty,
\end{equation}
which gives the first equality. Inequality $(a)$ follows directly from Proposition~\ref{CMC}. Equality $(b)$ is a direct consequence of \eqref{feqg} which gives that $\mathop{\mathbb{E}}\nolimits \mathcal{M}_T=1.$
Now, to obtain the lower bound
we must take
$$
\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,
$$
or equivalently
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To finish the proof we just replace $Y_t$ by $\nu_t+A_t \pi_{t-1}(X_{t})$. Thus in the last exponential term we find:
\begin{multline*}
\exp\left\{ - \frac{\nu_t^2 A_t^2 (\gamma_t-\overline{\gamma}_t)} {2(1+A_t^2 \overline{\gamma}_t)(1+A_t^2 \gamma_t)} + \frac{Z_{t}^{h}-\pi_{t-1}(X_{t})}{1+A_{t}^2\overline{\gamma}_t}A_t \nu_t \right. -
\\ - \left.\frac{1}{2}\cdot \frac{A_t^2}{1+A_t^2 \overline{\gamma}_t} (Z_{t}^h-\pi_{t-1}(X_{t}))^2 \right\},
\end{multline*}
which gives the Proposition.
\begin{rem}\label{probinterp}
\begin{enumerate}
\item
Note that now the probabilistic interpretation of the ingredients $\overline{\gamma}_t$ and $Z_t^{h}$ is clarified for \textbf{negative} $\mu$. Namely,
$\overline{\gamma}_t=\mathop{\mathbb{E}}\nolimits (X_t-\overline{\pi}_{t-1}(X_t))^2,$
and $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$, but when
$\mu$ is \textbf{positive}, there is no such connection anymore.
\item Observe that actually $\overline{\pi}_{t-1}(X_t)$ and $\overline{\gamma}_{_{X\xi}}(t)$ are $\bar{{\cal Y}}_{t-1}$-measurable, but the difference $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ is ${\cal Y}_{t-1}$ measurable.
\end{enumerate}
\end{rem}
\paragraph{Proof of Theorem \ref{LEGsol}} The statement of Theorem
\ref{LEGsol} is the direct consequence of
Proposition \ref{CMC}. Indeed, we claim that the following chain of inequalities holds for any $h:\,h_t\in {\cal Y}_{t}, t\ge 1$ :
$$
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
$$
$$
=\mathop{\mathbb{E}}\nolimits\left[\mathop{\mathbb{E}}\nolimits\mu \left(\left.\exp\left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\} \right/ {{\cal Y}}_{T}\right)\right]
$$
$$
=\mu\mathop{\mathbb{E}}\nolimits\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_t - \frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t } \right]^{2}\right\} \times \mathcal{M}_T,
$$
$$
\stackrel{(a)}{\ge}\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \mu \mathop{\mathbb{E}}\nolimits \mathcal{M}_T
$$
$$
\stackrel{(b)}{=} \mu \prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} .
$$
Of course under condition $(C_{\mu})$, since the term in the last line is finite, it is sufficient to consider the case:
\begin{equation}\label{expfin}
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
< \infty,
\end{equation}
which gives the first equality. Inequality $(a)$ follows directly from Proposition~\ref{CMC}. Equality $(b)$ is a direct consequence of \eqref{feqg} which gives that $\mathop{\mathbb{E}}\nolimits \mathcal{M}_T=1.$
Now, to obtain the lower bound
we must take
$$
\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,
$$
or equivalently
$$
\overline{h}_{t}= Z_{t}^{\overline{h}} + \frac{A_t \overline{\gamma}_{t}}{1+A_t^2 \overline{\gamma}_{t}} (Y_t - A_t Z_{t}^{\overline{h}}),\, t\ge 1,
$$
where $Z^{h}$ is the solution
of equation (\ref{eqzh}), which means that
$$
Z_{t}^{\overline{h}} =m_t +\sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l) A_l}{1+A_l^2 \overline{\gamma}_l} [Y_l -A_l Z_{l}^{\overline{h}}],
$$
and hence
$$
\overline{h}_{t}= m_t + \sum_{l=1}^{t} \frac{\overline{\gamma}(t,l) A_l}{1+A_l^2 \overline{\gamma}_l} [Y_l -A_l Z_{l}^{\overline{h}}]=m_t+\sum_{l=1}^{t} A_l \overline{\gamma}(t,l) (Y_l -A_l \overline{h}_l ).
$$
Thus
$\bar{h}$ is the unique solution of equation \eqref{hbar}.
Finally for $\bar{h}$ the lower bound is attained.
\begin{rem}\label{probinterp'}
\begin{enumerate}
\item It is worth emphasizing that the process $\displaystyle{\tilde{Z}^{h}_{t}=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t }}$ is the solution of the following recursive equation:
\begin{equation}\label{zhtild}
\tilde{Z}^{h}_{t}=m_t -\sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{\mu Q_l}{1+S_l \overline{\gamma}_l} (h_l - \tilde{Z}_{l}^h) + \sum_{l=1}^{t}\overline{\gamma}(t,l) A_l (Y_l - A_l \tilde{Z}_{l}^{h}),
\end{equation}
and hence the equality $\overline{h}_t =\displaystyle{\widetilde{Z}^{\overline{h}}_{t}} $ implies immediately the equation \eqref{eqzh} for $\overline{h}$.
This process $\tilde{Z}^{h}$ also has
a probabilistic interpretation as well as $\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$. This interpretation will be given in Section~\ref{interpret}.
\end{enumerate}
\end{rem}
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\subsection{Solution of the auxiliary filtering problems}\label{AFP}
Here, for an arbitrary Gaussian sequence $X$, we deal with the
one-step prediction
and filtering problems of the signals $X$
and $\xi$ given by \eqref{xi:eq} respectively from the observation of
$\bar{Y}=(Y,Y^{2})$ defined in \eqref{observ} and \eqref{Yaux}.
Actually, we follow the ideas proposed in our paper
\cite{mkalbmcr3}.
Recall that the solutions can
be reduced to equations for the conditional moments. The following
statement provides the equations for the characteristics which give
the solution
of the prediction problem and the equation for the other quantity
$\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ appearing in Proposition~\ref{p:CM} for $\mu=-1.$
\begin{theorem}\label{filter}
The conditional mean $\overline{\pi}_{t-1}(X_t)$ and the
variance of the one-step prediction error
$\overline{\gamma}_t=\mathop{\mathbb{E}}\nolimitsg[X_t-\overline{\pi}_{t-1}(X_t)]^{2}$ are given by the equations
\begin{multline}\label{eq1 filter}
\overline{\pi}_{t-1}(X_t)=m_{t}+\sum\limits_{s=1}^{t-1}
\frac{\overline{\gamma}(t,s)}{1+(A_s^2+Q_s)\overline{\gamma}_s}[A_s(Y_{s}-A_s\overline{\pi}_{s-1}(X_s)) \\
+Q_s(Y_s^2 - Q_s(\overline{\pi}_{s-1}(X_s)-h_s)]\,,\quad t\geq 1,
\end{multline}
\begin{equation}\label{eq2 filter}
\overline{\gamma}_t=\overline{\gamma}(t,t)\,,\quad t\geq 1\,.
\end{equation}
where $\overline{\gamma}=(\overline{\gamma}(t,s),\, 1\le s \le t)$ is the unique solution of equation \eqref{GAMMABAR}.
Moreover, with $\overline{\gamma}_{_{X\xi}}(t)$ defined by (\ref{def gamma Xxi}), the difference $\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ is the
solution $Z^h_t$ of equation (\ref{eqzh}).
\end{theorem}
\paragraph{Proof} Note that since $h_t \in {\cal Y}_t$ and the joint
distribution of $(X_{r},Y_{s}, Y_s^2+Q_s h_s)$ for any $r\,,s$ is Gaussian, we
can apply the Note following
Theorem 13.1 in \cite{lipshi1bis}. For any
$k\le t$ we can write
\begin{equation}\label{piX}
\left\{
\begin{array}{l}
\overline{\pi}_k(X_t)=\overline{\pi}_{k-1}(X_t) + [\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_k)]^{\prime} \mathop{\mathrm{var}}\nolimits(\overline{\nu}_k)^{-1} \overline{\nu}_k,
\\
\overline{\pi}_{0}(X_{t})=m_{t},
\end{array}
\right.
\end{equation}
where
$$
\overline{\nu}_{k}=\bar{Y}_{k}-\mathop{\mathbb{E}}\nolimitsg(\bar{Y}_{k}/\overline{{\cal Y}}_{k-1})=\left(
\begin{array}{c}
Y_k-A_k\overline{\pi}_{k-1}(X_k) \\
Y_k^2 +Q_kh_k - Q_k\overline{\pi}_{k-1}(X_k)
\end{array}
\right)
$$
is the innovation with covariance matrices
\begin{equation}\label{varnu}
\mathop{\mathrm{var}}\nolimits(\overline{\nu}_k)=\left(
\begin{array}{cc}
1+A_k^2\overline{\gamma}_k & A_kQ_k\overline{\gamma}_k\\
A_kQ_k\overline{\gamma}_k & Q_k+Q_k^2\overline{\gamma}_k
\end{array}
\right),
\end{equation}
and
\begin{equation}\label{covarXnu}
\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_k) = \overline{\gamma} (t,k) \left( \begin{array}{c} \! A_k \! \\ \! Q_k \! \end{array} \right),
\end{equation}
with
\begin{equation}\label{def gamma t s}
\overline{\gamma}(t,k)=\mathop{\mathbb{E}}\nolimitsg (X_t-\overline{\pi}_{k-1}(X_t))(X_{k}-\overline{\pi}_{k-1}(X_k))\,.
\end{equation}
By the definition (\ref{def gamma t s}), we see for $k=t$ that the
variance $\overline{\gamma}_t$ is given by (\ref{eq2 filter}). Now,
equality (\ref{piX}) implies
\begin{multline}\label{pikt}
\overline{\pi}_k(X_t)=m_t+\sum_{l=1}^k \overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l=
\\
=m_{t}+\sum\limits_{s=1}^{k}
\frac{\overline{\gamma}(t,s)}{1+(A_s^2+Q_s)\overline{\gamma}_s}[A_s(Y_{s}-A_s\overline{\pi}_{s-1}(X_s))+
\\
+Q_s(Y_s^2 - Q_s(\overline{\pi}_{s-1}(X_s)-h_s)],
\end{multline}
and putting $k=t-1$ we get nothing but equation (\ref{eq1
filter}). Concerning the solution of the one-step prediction
problem, it just remains to show that the covariance $\overline{\gamma}(t,s)$
satisfies equation (\ref{GAMMABAR}).
Let us define
$$
\delta_{X} (t,l) = X_t - \overline{\pi}_l (X_t)\,.
$$
According to (\ref{piX}) we can write
$$
\delta_X (t,l) = \delta_X (t,l-1) - \overline{\gamma}(t,l)\left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l,
$$
and so
\begin{multline*}
\mathop{\mathbb{E}}\nolimits \delta_X(t_1, l) \delta_X(t_2,l) = \mathop{\mathbb{E}}\nolimits \delta_X(t_1, l-1) \delta_X(t_2, l-1) -
\\
-\overline{\gamma}(t_1,l) \overline{\gamma}(t_2,l) \left({A_l \atop Q_l}\right)^{\prime} \mathop{\mathrm{var}}\nolimits(\bar{\nu}_l)^{-1} \left({A_l \atop Q_l}\right),
\end{multline*}
or
\begin{multline}\label{deltadelta}
\mathop{\mathbb{E}}\nolimitsg \delta_{X}(t^1,l)\delta_{X}(t^2,l)=\mathop{\mathbb{E}}\nolimitsg
\delta_{X}(t^1,0)\delta_{X}(t^2,0)- \\
-\sum\limits_{r=1}^{l}\overline{\gamma}(t,r) \overline{\gamma}(s,r) \frac{A_r^2+Q_r}{1+ (A_r^2 +Q_r ) \overline{\gamma}_{r}}.
\end{multline}
Taking $t^1=t\,, t^2=s\,, l= s-1$ in (\ref{deltadelta}), it is
readily seen that
equation (\ref{GAMMABAR}) holds for $\overline{\gamma}(t,s)$.
Now we analyze the difference $\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$.
Using the representation $\xi_t=\sum_{s=1}^t (X_s-h_s) Y_s^2$
we can rewrite $\overline{\pi}_{t-1}(\xi_{t-1})$ in the following form
$$
\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (\pi_{t-1}(X_s)-h_s) Y_s^2,
$$
which implies that
$$
\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (X_s - \overline{\pi}_{t-1}(X_s)) Y_s^2.
$$
So we have
\begin{multline}\label{ga X xi}
\overline{\gamma}_{_{X\xi}}(t) = \sum_{s=1}^{t-1} \overline{\pi}_{t-1} [(X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))] Y_s^2 =
\\
= \sum_{s=1}^{t-1} \mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t)) Y_s^2
= \sum_{s=1}^{t-1} \widetilde{\gamma}(t,s) Y_s^2,
\end{multline}
where
\begin{equation}\label{ad gamma }
\widetilde{\gamma}(t,s)=\mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))= \overline{\gamma}(s,t).
\end{equation}
Using the definitions (\ref{def gamma t s}) and (\ref{ad gamma })
we can write
$$
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) = - \mathop{\mathbb{E}}\nolimits X_t (\overline{\pi}_{t-1}(X_s) - \overline{\pi}_{s-1}(X_s)).
$$
Again, applying the Note following Theorem 13.1 in \cite{lipshi1bis}, we
can write also
$$
\overline{\pi}_{l}(X_{r})= \overline{\pi}_{l-1}(X_{r})+\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l.
$$
This means that
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})= \sum\limits_{l=r}^{t-1}\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l\,,
$$
or equivalently
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,t) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}
\overline{\nu}_l\,.
$$
Then, multiplying by $X_t$ and taking expectations in both sides, we get
\begin{multline*}
\mathop{\mathbb{E}}\nolimitsg X_{t}
(\pi_{t-1}(X_{r})-\pi_{r-1}(X_{r}))
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,r) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_l)=
\\
=\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{multline*}
Hence we have proved the following relation
\begin{equation}\label{dif gamma}
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) =-\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{equation}
Now we can show that the difference $Z^h_{t}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$
satisfies the equation (\ref{eqzh}). Using (\ref{pikt}) and (\ref{ga X xi}),
we can write
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Now we analyze the difference $\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$.
Using the representation $\xi_t=\sum_{s=1}^t (X_s-h_s) Y_s^2$
we can rewrite $\overline{\pi}_{t-1}(\xi_{t-1})$ in the following form
$$
\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (\pi_{t-1}(X_s)-h_s) Y_s^2,
$$
which implies that
$$
\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (X_s - \overline{\pi}_{t-1}(X_s)) Y_s^2.
$$
So we have
\begin{multline}\label{ga X xi}
\overline{\gamma}_{_{X\xi}}(t) = \sum_{s=1}^{t-1} \overline{\pi}_{t-1} [(X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))] Y_s^2 =
\\
= \sum_{s=1}^{t-1} \mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t)) Y_s^2
= \sum_{s=1}^{t-1} \widetilde{\gamma}(t,s) Y_s^2,
\end{multline}
where
\begin{equation}\label{ad gamma }
\widetilde{\gamma}(t,s)=\mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))= \overline{\gamma}(s,t).
\end{equation}
Using the definitions (\ref{def gamma t s}) and (\ref{ad gamma })
we can write
$$
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) = - \mathop{\mathbb{E}}\nolimits X_t (\overline{\pi}_{t-1}(X_s) - \overline{\pi}_{s-1}(X_s)).
$$
Again, applying the Note following Theorem 13.1 in \cite{lipshi1bis}, we
can write also
$$
\overline{\pi}_{l}(X_{r})= \overline{\pi}_{l-1}(X_{r})+\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l.
$$
This means that
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})= \sum\limits_{l=r}^{t-1}\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l\,,
$$
or equivalently
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,t) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}
\overline{\nu}_l\,.
$$
Then, multiplying by $X_t$ and taking expectations in both sides, we get
\begin{multline*}
\mathop{\mathbb{E}}\nolimitsg X_{t}
(\pi_{t-1}(X_{r})-\pi_{r-1}(X_{r}))
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,r) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_l)=
\\
=\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{multline*}
Hence we have proved the following relation
\begin{equation}\label{dif gamma}
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) =-\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{equation}
Now we can show that the difference $Z^h_{t}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$
satisfies the equation (\ref{eqzh}). Using (\ref{pikt}) and (\ref{ga X xi}),
we can write
\begin{multline}\label{zhbeg}
Z^h_{t} = m_t+\sum_{l=1}^{t-1} \overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l - \sum_{s=1}^{t-1} \widetilde{\gamma}(t,s) Y_s^2 =
\\
= m_t +\sum_{l=1}^{t-1} \frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l}(Y_l - A_l \overline{\pi}_{l-1}(X_l)) +
\\
+
\sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l}(Y_l^2-Q_l (\overline{\pi}_{l-1}(X_l)-h_l))
-\sum_{l=1}^{t-1} \widetilde{\gamma}(t,l) Y_l^2 =
\\
= m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Q_l h_l - \\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} \overline{\pi}_{l-1}(X_l) +
\\
+ \sum_{l=1}^{t-1} [\frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} - \widetilde{\gamma} (t,l)] Y_l^2.
\end{multline}
Now we can rewrite the last term in \eqref{zhbeg} using the equality \eqref{dif gamma}. We have
\begin{multline}\label{gammause}
\sum_{l=1}^{t-1} [\frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} - \widetilde{\gamma} (t,l)] Y_l^2= \sum_{l=1}^{t-1} \overline{\gamma}(t,l) (\frac{1} {1+(A_l^2 +Q_l)\overline{\gamma}_l}-1) Y_l^2 +
\\
+ \sum_{l=1}^{t-1} \sum_{r=l}^{t-1} \overline{\gamma}(t,r) \widetilde{\gamma}(r,l)\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r} Y_l^2 = \\
= \sum_{r=1}^{t-1} \overline{\gamma}(t,r)
\left[\sum_{l=1}^{r-1} \widetilde{\gamma}(r,l) Y_l^2 \right]
\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r}=
\\
=\sum_{r=1}^{t-1} \overline{\gamma}(t,r)\overline{\gamma}_{_{X\xi}}(r)\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r}\,,
\end{multline}
where in the last step we have used equality (\ref{ga X xi}).
Finally \eqref{zhbeg}-\eqref{gammause} imply:
\begin{multline*}
Z^h_{t} = m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{Q_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} h_l -
\\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l}[ \overline{\pi}_{l-1}(X_l)-\overline{\gamma}_{_{X\xi}}(l)]=
\\
m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{Q_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} h_l -
\\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l}Z^h_{l}\,,
\end{multline*}
which is nothing else but equation (\ref{eqzh}) with $\mu=-1$.
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\section{Particular cases and applications}\label{PC}
Here we deal with some specific cases where the results can be further elaborated. For two examples we can apply directly Theorem~\ref{LEGsol} and moreover
the special structure of the covariances allows to simplify the answer.
\subsection{LEG filtering of Gauss-Markov sequences }\label{GMS}
In this part we concentrate on the case of a
Gaussian AR(1) process $X$, {\em i.e.}, a Gauss-Markov process
driven by
\begin{equation}\label{model AR}
X_{t}= a_{t} X_{t-1}+D_t^\frac{1}{2}\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}\,,\; t\ge 1\,;
\quad X_{0}=x\,,
\end{equation}
where $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables and
$(D_t,\,t\ge 1)$ is a (deterministic)
sequence of real numbers such that $D_t\ge 0$ for $t\ge 1$. In this setting,
it is easy to check that the mean
and covariance functions of $X$ are given by
$$
m_{t}=[\prod_{u=1}^{t}a_u]x=\Lambda_{t}x\,;\quad
K(t,s)=[\prod_{u=s+1}^{t}a_u]k_s=\frac{\Lambda_{t}}{\Lambda_{s}}k_s\,,\;1\le s\le t\,,
$$
where $\Lambda_{t}=\prod\limits_{u=1}^{t}a_u$ and
$$
k_t=a_t^2k_{t-1} +D_t,\, t\ge 1,\,k_0=0.
$$
Suppose that the following the Riccati type equation
\begin{equation}\label{ricmar}
\overline{\gamma}_s=D_s+\frac{a_{s}^2 \overline{\gamma}_{s-1}}{1+(A_{s-1}^2-\mu Q_{s-1})\overline{\gamma}_{s-1}},\, s\ge 1,\,\overline{\gamma}_{0}=0,
\end{equation}
has a unique nonnegative solution.
From the classical filtering theory it is
well-known that (for $\mu<0$ ) $\overline{\gamma}_s$ is nothing but the variance of the error of the one-step prediction problem
of the signal $X$ given by the auxiliary observation $\bar{Y}$ defined by equations \eqref{observ} and \eqref{Yaux}. Then, it is readily
seen that the function $\overline{\gamma}(t,s)$, where
$\displaystyle{\overline{\gamma}(t,s)=\frac{\Lambda_{t}}{\Lambda_{s}}\overline{\gamma}_s}$ is the
solution of equation (\ref{GAMMABAR}) and that moreover equation
\eqref{hbar} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation}\label{rsmkk}
\overline{h}_t=\frac{a_t}{1+A_t^2\overline{\gamma}_t}\overline{h}_{t-1} +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t}Y_t, \, t\ge 1,\, \overline{h}_0=x,
\end{equation}
or, equivalently:
\begin{equation*}
\overline{h}_t=a_t \overline{h}_{t-1} +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t} [Y_t-a_tA_t \overline{h}_{t-1}], \, t\ge 1,\, \overline{h}_0=x.
\end{equation*}
Actually equation \eqref{rsmkk} can also be obtained directly from the general filtering theory (for $\mu=-1$ and replacing $Q$ by $-\mu Q$).
For arbitrary $(h_{t} \in {\cal Y}_{t}, t\ge 1)$ the Note following
Theorem 13.1 in \cite{lipshi1bis} gives the equation for $Z^{h}$:
\begin{multline*}
Z^{h}_{t}=a_t Z^{h}_{t-1} +a_t \overline{\gamma}_t\frac{Q_{t-1} }{1+S_{t-1}\overline{\gamma}_t} [h_{t-1}-Z^h_{t-1}] +
\\
+a_t \overline{\gamma}_t\frac{A_{t-1} }{1+S_{t-1}\overline{\gamma}_t} [Y_{t-1}-A_{t-1}Z^h_{t-1}], \, t\ge 1,\, Z^h_0=x.
\end{multline*}
Hence, again the solution
$\displaystyle{\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,
}$ of the LEG filtering problem (\ref{LEGdef}) is given by \eqref{rsmkk}.
Let us emphasize that these equations are nothing but those given in Speyer \textit{et al.}
\cite{speyer}.
It is interesting to note that in the case $a_t=0$ (i.i.d. signal) the solution of the LEG filtering problem is
nothing else but the solution of the risk neutral filtering problem \textit{i.e.} $\overline{h}_t=\pi_t(X_t)$.
\subsection{LEG filtering of moving averages of order 1}\label{MA1X}
Here we consider the case of a MA(1) process, {\em i.e}, a non
Markovian process $X$ defined by
$$
X_{t} = \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t} + \lambda \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t-1}\,; t\ge 1\,,
$$
where $( \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_0, \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_1,\dots)$~is a sequence of i.i.d. standard
Gaussian variables and $\lambda$ is a real number. Of course
$X$ is centered and has the
covariance function
$K(t,s)= 1+\lambda^{2}$ if $s=t$, $\lambda$ if $s=t-1$ and
$0$ if $s < t-1$.
In order to solve equation (\ref{GAMMABAR}) we can take
$$
\overline{\gamma}(t,s) =0\,,\; s < t-1\,; \quad \overline{\gamma}(t,t-1) = \lambda\,,\;
t\ge 1\,,
$$
and $\overline{\gamma}(t,t)=\overline{\gamma}_{t}$ where $\overline{\gamma}_{t}$ is
the solution of the equation:
$$
\overline{\gamma}_{t} =1+\lambda^{2}
- \lambda \frac{A_{t-1}^2-\mu Q_{t-1}}{1+(A_{t-1}^2-\mu Q_{t-1})\overline{\gamma}_{t-1}}\,,\;t\ge 1\,;\quad \overline{\gamma}_0=1+\lambda^{2},
$$
provided that this equation has a unique nonnegative solution.
Moreover equation
\eqref{hbar} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation*}
\overline{h}_t=\lambda\frac{A_{t-1}}{1+A_t^2\overline{\gamma}_t}[Y_{t-1}-A_{t-1}\overline{h}_{t-1}] +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t}Y_t, \, t\ge 1,\, \overline{h}_0=0.
\end{equation*}
Again, it is interesting to note that for $\lambda=0$ (i.i.d. signal) the solution of LEG filtering problem is
nothing else but the solution of the risk neutral filtering problem \textit{i.e.} $\overline{h}_t=\pi_t(X_t)$.
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\section{LEG and RS filtering problems}\label{disc}
Here, at first we show that actually the LEG and RS filtering problems have the same solution. Then we give an example which shows that in a more general context similar problems may have different solutions.
\subsection{Equivalence of LEG and RS filtering problems}
Let $\bar{h}=(\overline{h}_s)_{s \ge 1}$ be the solution of the LEG filtering problem (\ref{LEGdef})
given by equation \eqref{hbar}.
For any fixed $t \le T$, let us denote by $\hat{g}_{t}:$
\begin{multline*}
\hat{g}_{_t}=
\displaystyle{\arg\min_{g \in {\cal Y}_t} }\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}(X_{t}-g)^{2}Q_{t}}
+\left.\displaystyle{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-\bar{h}(s))^{2}Q_{s} }\Big\} \right/
{\cal Y}_t\Big],
\end{multline*}
where $g\in {\cal Y}_t$ means that $g$ is a ${\cal Y}_t$-measurable variable.
It follows directly from
Proposition~\ref{p:CM}
that, provided that $\displaystyle{1 + S_t\overline{\gamma}_t}> 0$, the equality
$\displaystyle{\hat{g}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1}$ holds.
Since it was noted in the proof of Theorem~\ref{LEGsol} that $\displaystyle{\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,}$
hence we have also $\hat{g}_{t}=\bar{h}_t$. It means that for $ t\ge 1 $
the solution $\bar{h}$ of the LEG filtering problem satisfies the following recursive
equation:
\begin{multline}\label{rde}
\hat{g}_{_t}=
\displaystyle{\arg\min_{g \in {\cal Y}_t} }\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}(X_{t}-g)^{2}Q_{t}}
+\left.\displaystyle{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-\bar{h}(s))^{2}Q_{s} }\Big\} \right/
{\cal Y}_t\Big].
\end{multline}
Indeed, in the literature, the recursion (\ref{rde}) is the basic
\textbf{definition} of the so-called risk-sensitive (RS)
filtering problem which was introduced in \cite{elliott3}. Therefore we have also proved the following
statement
\begin{theorem}\label{RS}
Assume that the condition $(C_{\mu})$ is
satisfied. Let $\overline{h}=(\overline{h}_t)_{t\ge 1}$ be the unique solution of equation
(\ref{hbar}), \textit{i.e.}, $\bar{h}$ is the solution of the LEG
filtering problem \eqref{LEGdef}.
Then $\bar{h}$ is the solution of the RS filtering problem
\eqref{rde}.
\end{theorem}
\subsection{Discrepancy between LEG and RS type filtering problems: an example }\label{EX}
Actually, we did not find in the literature any trace of the
discussion
about the relationship between the LEG
filtering problem (\ref{LEGdef}) and the RS filtering
problem (\ref{rde}) even in a Gauss-Markov case. As a complement to our observation that these two problems
have the same solution, we propose an example to show that in a bit more general setting,
two similar problems may have different solutions.
For given positive symmetric deterministic $2\times 2$ matrices
$\Lambda_s,1\le s\le T$, let us set $\Phi_{t}(h)=(X_{t} \, h_t)\Lambda_{t}\left(
\begin{array}{c}
X_{t} \\
h_t \\
\end{array}
\right)$. We can define $\bar{h}_{t}\in {{\cal Y}}_{t},\, t\ge 1$ as a solution of a
\textit{LEG type filtering problem} :
\begin{equation}\label{defrssex}
\overline{h}= \arg\min_{h_{t}\in {\cal Y}_{t},\, t\ge 1}\mathop{\mathbb{E}}\nolimitsg \left[\mu \exp\left\{\frac{\mu}{2}
\sum_1^T \Phi_{s}(h)\right\}\right].
\end{equation}
We can also define
$\hat{h}$ as the solution of
the following recursive equation (\textit{RS type filtering problem}):
\begin{equation}\label{rdeex}
\hat{h}_{t}= \displaystyle{\arg\min_{g \in {\cal Y}_t}
}\,\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}\Phi_{t}(g)\,}+\left.\displaystyle{\frac{\mu}{2} \sum_{1}^{t-1}
\Phi_{s}(\hat{h})\, }\Big\} \right/ {\cal Y}_t\Big].
\end{equation}
The question which we discuss now is the following: does the equality
$\bar{h}=\hat{h}$ hold?
As we have just proved, the answer is positive for singular matrices
$\Lambda$, namely, when
$\Lambda_{11}=\Lambda_{22}=-\Lambda_{12}=Q$.
But in the general situation the answer may be negative. Actually it is sufficient to consider the following example:
$\Lambda =\left(
\begin{array}{cc}
2 & -1 \\
-1 & 1 \\
\end{array}
\right),\, A_{t}=1,\,\mu=-1 $ and $X_{t}= X_{t-1}+\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}$, where $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$
is a sequence of i.i.d.
standard Gaussian random variables. Even in this Markov
case $\hat{h}\ne \overline{h} $. More explicitly let us introduce the new probability measure $\hat{\Pg}:$
$$
\frac{d\hat{\Pg}}{d\Pg}= \frac{\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right]}{\mathop{\mathbb{E}}\nolimitsg\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right]}.
$$
One can check that with respect to $\hat{\Pg}$ the observation model $(X_t,Y_t)_{t \ge 1}$ can be written in the following form:
$$
X_t= a_tX_{t-1} + D_t^\frac{1}{2}\hat{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}\,,\; t\ge 1\,;
\quad X_{0}=x\,,
$$
$$
Y_t=X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t\,,
$$
where $(\hat\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d.
standard Gaussian random variables independent of the sequence $\mathop{\mathrm{var}}\nolimitsepsilon$,
$$
a_t=D_t=\frac{1}{1+ \Gamma(T,t)}\,,
$$
and $\Gamma(T,\cdot)$ is the solution of the backward Riccati equation
$$
\Gamma(T,t) = 1 +\frac{\Gamma(T,t+1)}{1+\Gamma(T,t+1)},\,\Gamma(T,T)=0.
$$
It can be checked that
$$
\Gamma(T,t)=10\frac{\lambda^{T}-\lambda^{t}}{(1-\sqrt{5})\lambda^{T}-(1+\sqrt{5})\lambda^{t}},\, \lambda = \frac{(3-\sqrt{5})}{(3+\sqrt{5})}.
$$
Indeed to explain this change of the observation model it is sufficient to calculate the conditional characteristic function:
$$
\hat{\mathop{\mathbb{E}}\nolimitsg}\left[\left.\exp (i\lambda X_t)\right/{\cal
X}_{t-1}\right]=\frac{\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left[i\lambda X_t-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right] \right/{\cal
X}_{t-1}\right]}{\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right] \right/{\cal
X}_{t-1}\right]},
$$
where ${\cal
X}_{t-1}$ is the
$\sigma$-field
${\cal
X}_{t-1}=\sigma(\{X_s\, ,\, 1\leq s\leq t-1\})$.
But it follows directly from the equation (19)-(20) in \cite{mkalbmcr3} and from \eqref{feqg}
that
$$\hat{\mathop{\mathbb{E}}\nolimitsg}\left[\left.\exp (i\lambda X_t)\right/{\cal
X}_{t-1}\right]=\exp\left\{\frac{i\lambda}{1+ \Gamma(T,t)}X_{t-1} -\frac{\lambda^{2}}{2(1+ \Gamma(T,t))}\right\}.
$$
Since the density $\displaystyle{\frac{d\widehat{\Pg}}{d\Pg}}$ does not depend on $h$, the initial LEG filtering problem \eqref{defrssex} can be rewritten as:
$$
\overline{h}= \arg\min_{h_{t}\in {\cal Y}_{t},\, 1\le t\le T}\hat{\mathop{\mathbb{E}}\nolimitsg} \left[- \exp\left\{-\frac{1}{2}
\sum_1^T (X_s-h_s)^{2}\right\}\right].
$$
Hence we can apply Theorem~\ref{LEGsol} or in particular \eqref{ricmar} and \eqref{rsmkk}. Clearly, $\overline{h}$ depends on $T$ and $\hat{h}$ does not depend on $T$ by the definition. A bit more explicitly we have for example for $t=1$:
$
\displaystyle{\overline{h}_{1}=\frac{1+\Gamma(T,1)}{2+\Gamma(T,1)}Y_1}
$
and obviously
$
\displaystyle{\hat{h}_{1}=\frac{\pi_{1}(X_1)}{1+\gamma_{1}}=\frac{1}{4}Y_1}
$
and clearly they are different.
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\section{Information state, interpretation}\label{interpret}
In this section we discuss the probabilistic interpretation of the ingredients of the ``information state'' which was introduced
in the context of RS filtering and LEG control problems.
By the definition, the ``information state'' contains all the information needed to describe the solution of the concerned optimization problem. In particular it takes into account the cost function but not only estimates of the signal and it should
give the total information about the model states available in the measurement.
\textit{Risk-Sensitive Filtering}
\noindent
In the context of the RS filtering problem the definition of the information state can be found for example in \cite{elliott1}.
It is the density $\lambda_{t}$, with respect to the Lebesgue measure, of the non normalized random measure
$\omega_{t}$:
\begin{equation}\label{mescond}
\omega_{t}(dx)=\displaystyle{\mathop{\mathbb{E}}\nolimitsg\left[\left.\mathbb{I}(X_t \in dx)\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-h(s))^{2}Q_{s}\right\} \right/ {\cal
Y}_t\right]},
\end{equation}
where $h_t\in {\cal Y}_t, t\ge 1 $
and the observation $Y$ is defined by the equation \eqref{observ}.\\
In a classical Gauss-Markov setting, an explicit representation of $\lambda_{t}$ can be obtained as the solution of some
recurrence equation (see, \textit{e.g.},
\cite{collings}).\\
We claim that for a general Gaussian signal $X$ the density $\lambda_{t}$
satisfies the following equality:
\begin{multline}\label{cdensg}
\lambda_{t}(x) =
\displaystyle{\frac{1}{\sqrt{2\pi \widetilde{\gamma}_t}}
\exp\left\{-\frac{(x-\widetilde{Z}_{t}^h)^{2}}{2\widetilde{\gamma}_{t}}
\right\}}\times
\\
\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r} \times \left[ h_r -\widetilde{Z}_{t}^h \right]^{2}\right\} \times \mathcal{M}_t,
\end{multline}
where $\displaystyle{ \widetilde{Z}_{t}^h=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+A_t^2\overline{\gamma}_t}}$ is the solution of the equation \eqref{zhtild}, $\displaystyle{ \widetilde{\gamma}_{t}= \frac{\overline{\gamma}_{t}}{1+A_t^2\overline{\gamma}_t}}$, $\overline{\gamma}, Z^h$ are the solutions of
equations (\ref{GAMMABAR}) and (\ref{eqzh}) respectively and the martingale $(\mathcal{M}_t)_{\ge 1}$ is defined by \eqref{martin}.
Indeed, to prove (\ref{cdensg}) it is sufficient to write the following:
\begin{equation}\label{bayesford}
\omega_{t}(dx) = \frac{\mathop{\mathbb{E}}\nolimitsg[\mathbb{I}(X_t \in dx)\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}{\mathop{\mathbb{E}}\nolimitsg[\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-h(s))^{2}Q_{s}\right\}\right/ {\cal
Y}_t\right],
\end{equation}
where
$\sigma$-field
${\overline{{\cal Y}}}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Again, conditionally Gaussian
properties of the pair $(X,\xi)$ imply that
\begin{equation*}
\frac{\mathop{\mathbb{E}}\nolimitsg \left[\mathbb{I}(X_{t} \in dx)\exp\left\{-\xi_{t} \right\}/
{\cal \overline{Y}}_{t,t-1} \right]}{\mathop{\mathbb{E}}\nolimitsg[\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}=
\displaystyle{[2\pi\widetilde{\gamma}_{t}]^{-\frac{1}{2}}}
\end{equation*}
\begin{equation}\label{gausden}
\times \exp \displaystyle{
\left\{-\frac{1}{2}(x-\widetilde{Z}^{h}_{t})^{2}\widetilde{\gamma}^{-1}_t
\right\}\,dx},
\end{equation}
where $\displaystyle{\widetilde{Z}^{h}_{t}=\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}]- \mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t,t-1}]}$ and $\displaystyle{\widetilde{\gamma}_{t}=\mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])]^{2}}$.
Now the desired equality \eqref{bayesford} follows directly from Proposition~\ref{p:CM}.\\
It is worth emphasizing that (for negative $\mu$) now we know the probabilistic interpretation of the involved processes
$(Z^h,\widetilde{Z}^{h}, \,\bar{\gamma},\widetilde{\gamma})$. Actually we have proved that $Z^{h}$
is the difference $\bar{\pi}_{t}(X)-\bar{\gamma}_{_{X\xi}}(t)$ and
$\bar{\gamma}$ is nothing but the covariance of the filtering
error of $X$ in view of auxiliary observations $\bar{Y}$.\\ For the pair $(\widetilde{Z}^{h}, \,\widetilde{\gamma})$ we have the same relations but with respect to the $\sigma$-field
${\cal
Y}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Of course, after a simple integration of $\lambda_{t}$, formula (\ref{cdensg}) gives
Proposition~\ref{CMC} and therefore the solution of the LEG and RS filtering problems.
Let us also observe that the relations
$\displaystyle{\tilde{Z}^{h}_{t}=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t }},\,
\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$ which were announced in Remark~\ref{probinterp'} follow from the Note following
Theorem 13.1 in \cite{lipshi1bis}.
\textit{Linear Exponential Gaussian Control}
\noindent
In the context of the LEG control problem for a partially observed process, the information state is also defined (see, \textit{e.g.}, \cite{elliott1}) as the density $\lambda_{t}$, with respect to the Lebesgue measure,
of the non normalized random measure
$\omega_{t}$:
\begin{equation}\label{mescondcon}
\omega_{t}(dx)=\displaystyle{\mathop{\mathbb{E}}\nolimitsg\left[\left.\mathbb{I}(X_t \in dx)\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
X_{s}^{2}Q_{s} \right\} \right/ {\cal
Y}_t\right]},
\end{equation}
where $X$ is the controlled state governed by the equation:
\begin{equation}\label{SDECONT}
X_t=a_tX_{t-1}+b_tu_t +\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t\,,\;t\geq 1\,;\; X_0=0\,,
\end{equation}
$( \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t \ge 1}$ is a sequence of i.i.d. standard
Gaussian variables
and $u_{t}\in {\cal Y}_{t-1} $ corresponding to the available observation $Y$ defined by the equation \eqref{observ}.\\
By the same way that we have just explained, for the conditionally Gaussian pair $(X,Y)$, one can check that the density $\lambda_{t}$
satisfies the following equality:
\begin{multline}\label{cdensgcon}
\lambda_{t}(x) =
\displaystyle{\frac{1}{\sqrt{2\pi \widetilde{\gamma}_t}}
\exp\left\{-\frac{(x-\widetilde{Z}_{t})^{2}}{2\widetilde{\gamma}_{t}}
\right\}}\times
\\
\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r} \times \widetilde{Z}_{t}^{2}\right\} \times \mathcal{M}_t,
\end{multline}
where $\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$, $\bar{\gamma}$ is the solutions of
equation (\ref{GAMMABAR}), the martingale $(\mathcal{M}_t)_{\ge 1}$ is defined by \eqref{martin}
and $\widetilde{Z}$ is the solution of the equation
\begin{equation}\label{z:repres}
\widetilde{Z}_{t} = \frac{a_t}{1+ S_t \overline{\gamma}_t }\widetilde{Z}_{t-1}+\frac{b_t}{1+ S_t \overline{\gamma}_t }u_t
+ \bar{\gamma}_t A_tY_{t}.
\end{equation}
Actually it is the equation for the difference $\widetilde{Z}=\bar{\pi}_{t,t-1}(X)-\bar{\gamma}_{X\xi}(t,t-1)$, where the conditional expectations are taken with respect to the auxiliary observation process $\bar{Y}$ defined by the equations \eqref{observ} and \eqref{Yaux} with $h=0$.\\
Equality \eqref{cdensgcon} gives the possibility to rewrite the cost function in terms of the completely observable process $\widetilde{Z}$, namely:
\begin{equation}\nonumber
\begin{array}{ccl}
\mathop{\mathbb{E}}\nolimitsg\Big[
\exp\Big\{\displaystyle{\frac{\mu}{2} \sum_{s=1}^T
X_{s}^{2}Q_{s} }
\Big]
=\mathop{\mathbb{E}}\nolimitsg\Big\{ \mathop{\mathbb{E}}\nolimitsg\Big[\left.
\exp\Big\{\displaystyle{\frac{\mu}{2} \sum_{s=1}^T
X_{s}^{2}Q_{s} }\Big\} \right/
{\cal Y}_T\Big]\Big\}
\\
=\displaystyle{\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2}} \mathop{\mathbb{E}}\nolimitsg\Big[\displaystyle{\exp\left\{\frac{\mu}{2}\sum_{s=1}^T\widetilde{Z}_{s}^{2}\widetilde{Q}_{s}
\right\}}\Big]\times \mathcal{M}_T
\\
=\displaystyle{\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2}} \widetilde{\mathop{\mathbb{E}}\nolimitsg}\Big[\displaystyle{\exp\left\{\frac{\mu}{2}\sum_{s=1}^T\widetilde{Z}_{s}^{2}\widetilde{Q}_{s}
\right\}}\Big],
\end{array}
\end{equation}
where $\displaystyle{\widetilde{Q}_{r} = Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r}}$ and $\widetilde{\mathop{\mathbb{E}}\nolimitsg}$ stands for an expectation with respect to the new measure $\widetilde{\Pg}$ such that:
$$
\frac{d\widetilde{\Pg}}{d\Pg} =\mathcal{M}_T .
$$
With respect to this new measure the solution of equation \eqref{z:repres} can be represented as
\begin{equation}\label{z:repres'}
\widetilde{Z}_{t} = a_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t}\widetilde{Z}_{t-1}+b_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t}u_t
+ \frac{\bar{\gamma}_tA_t}{1+A_r^{2} \overline{\gamma}_r}\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_{t},
\end{equation}
where $( \bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t \ge 1}$ is a new sequence of i.i.d. standard
Gaussian variables.
Thus, the new process $\widetilde{Z}$ plays the role of the completely observed controlled state (see \cite{bensoussan} and \cite{elliott1}).
Now we emphasize that the probabilistic interpretation of the
``information state" $\widetilde{Z}$, used in \cite{elliott1}
is nothing but
$\widetilde{Z}_{t}=\bar{\pi}_{t,t-1}(X)-\bar{\gamma}_{X\xi}(t)$, where the conditional expectations are taken with respect to the auxiliary observation process $\bar{Y}$ defined by the equations \eqref{observ} and \eqref{Yaux} with $h=0$. Also, $\bar{\gamma}$ is the conditional covariance of $X$.
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\section{Complementary part - More general case}\label{complements}
In this section we analyze LEG and RS filtering problems in a more general contexts when we do not suppose a special structure of the observation sequence $(Y_t)_{t\ge 1}$. We suppose only that the process $(X_t,\,Y_t)_{t\ge 1}$ is Gaussian (even conditionally Gaussian). Our goal is to reduce LEG (RS) filtering problems to an auxiliary risk-neutral filtering problem.
First of all we fix $\mu=-1$ and we will find the probabilistic interpretation of the solution. After to find the solution for $\mu \ne -1$ we shall have only to replace $Q$ by $-\mu Q$ in the answer.
So, let $(Y^{2}_{t},\, \xi_{t})$ be defined by equations \eqref{Yaux} - \eqref{xi:eq} and let us denote by
\begin{equation}\label{Ztildainterpr}
\widetilde{Z}^{h}_{t}=\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}]- \mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t,t-1}],
\end{equation}
\begin{equation}
\widetilde{\gamma}_{t}=\mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])]^{2},
\end{equation}
where ${
\bar{{\cal Y}}}_{t,t-1}$ is the
$\sigma$-field
${
\bar{{\cal Y}}}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Again, let $\displaystyle {J_t= \exp\left\{-\frac{1}{2} \sum_{s=1}^t (X_s-h_s)^2 Q_s \right\}}$ and let us denote by $\mathcal{I}_t$ the conditional expectation
$\displaystyle {\mathcal{I}_t= \pi_t(J_t)}$, or
$$
\mathcal{I}_t= \mathop{\mathbb{E}}\nolimits \left(\left.\exp\left\{-\frac{1}{2} \sum_{s=1}^t (X_s-h_s)^2 Q_s \right\} \right/ {{\cal Y}}_{t}\right),
$$
where $h_s\in {\cal Y}_{s}, \, s\ge 1$.
We claim the following generalization of Proposition~\ref{p:CM}.
\begin{proposition}\label{p:CMG}
The following equality holds for any $T\ge 1$:
$$
\mathcal{I}_T=\prod_{t=1}^T \left[1+Q_t\widetilde{\gamma}_{t}\right]^{1/2} \times \exp\left\{-\frac{1}{2} \frac{Q_t}{1+ Q_t \widetilde{\gamma}_t} \times \left[ h_t -\widetilde{Z}^{h}_{t} \right]^{2}\right\} \times \mathcal{M}_T,
$$
where $(\mathcal{M}_T)_{T\ge 1}$ is a martingale defined by :
\begin{multline}\label{martin}
\mathcal{M}_T = \prod_{t=1}^T \left[\frac{\sigma_t^{2}}{\bar{\sigma}_t^{2} }\right]^{1/2} \exp\left\{ \frac{1}{2\sigma_t^{2}} \, (Y_t - \pi_{t-1}(Y_t) )^{2} -\frac{1}{2\bar{\sigma}_t^{2}} \, (Y_t -\bar{ V}_{t} )^{2}
\right\},
\end{multline}
where
$$
\sigma_t^{2}=\mathop{\mathbb{E}}\nolimitsg(Y_t-\pi_{t-1}(Y_t))^{2},\,
\bar{\sigma}_t^{2}=\mathop{\mathbb{E}}\nolimitsg(Y_t-\bar{\pi}_{t-1}(Y_t))^{2},
$$
$$
\bar{ V}_{t}=\bar{\pi}_{t-1}(Y_t) -\overline{\gamma}_{_{Y\xi}}(t),\, \overline{\gamma}_{_{Y\xi}}(t)=\mathop{\mathbb{E}}\nolimitsg[(Y_t-\bar{\pi}_{t-1}(Y_t))
(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t-1}].
$$
\end{proposition}
\paragraph{Proof}To prove Proposition~\ref{p:CMG}
let us again calculate the ratio
$$
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})}= \frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}\frac{\pi_{t}(J_{t-1})}{\pi_{t-1}(J_{t-1})}=
$$
$$
=\frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}\frac{\mathcal{M}_t}{\mathcal{M}_{t-1}}
$$
with a martingale $\mathcal{M}_t$ such that:
\begin{equation}\label{martgen}
\mathcal{M}_{t}=\prod_{s=1}^{t}\frac{\pi_{s}(J_{s-1})}{\pi_{s-1}(J_{s-1})}.
\end{equation}
The same arguments that we used in the proof of Proposition~\ref{p:CM} show that
$$
\frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}= \frac{\overline{\pi}_{t,t-1}(\exp \{-\frac{1}{2}Q_t (X_t-h_t)^2 - \xi_{t-1}\} )}{\overline{\pi}_{t,t-1}(\exp(-\xi_{t-1}))}
$$
$$
=(1+Q_t\widetilde{\gamma}(t))^{-1/2} \exp\left\{ - \frac{1}{2} \cdot\frac{Q_t}{1+Q_t\widetilde{\gamma}(t)}(\widetilde{Z}^{h}_{t}-h_t)^2 \right\}.
$$
To finish the proof we turn to the representation of the martingale $\mathcal{M}_t$ defined by \eqref{martgen}.
First of all we claim that
\begin{equation}\label{martbayes}
\frac{\mathcal{M}_{t}}{\mathcal{M}_{t-1}}=\left.\frac{\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))}{\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t},
\end{equation}
where $\widetilde{\pi}$ stands for the conditional expectation with respect to the measure $\widetilde{\Pg}$ such that
$\displaystyle{\frac{d \widetilde{\Pg}}{d \Pg}={\mathcal{M}_{T}}}$. Indeed, it is the direct consequence of the classical Bayes formula
$$
\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))=\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy){\mathcal{M}_{T}})}{\pi_{t-1}({\mathcal{M}_{t-1}})}
=\pi_{t-1}(\mathbb{I}(Y_t \in dy){\mathcal{M}_{t}}).
$$
To finish the proof it is sufficient to note that representations \eqref{martgen} and \eqref{martbayes} imply that
$$
\frac{\mathcal{M}_t}{\mathcal{M}_{t-1}}=\left.\frac{\pi_{t}(J_{t-1})}{\pi_{t-1}(J_{t-1})}=\left.\frac{\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))}{\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}=\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)\pi_{t}(J_{t-1}))}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}=
$$
$$
\left.\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)J_{t-1})}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}.
$$
Again, we can use the same arguments that we used in the proof of Proposition~\ref{p:CM}:
$$
\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)J_{t-1})}{\pi_{t-1}(J_{t-1})}=\frac{\overline{\pi}_{t-1}(\mathbb{I}(Y_t \in dy)\exp\{-\xi_{t-1}\})}{\overline{\pi}_{t-1}(\exp\{-\xi_{t-1}\})}=
$$
$$
=\frac{1}{\sqrt{2\pi \bar{\sigma}^{2}}}\exp\left(-\frac{(Y_t-\bar{V}_t )^{2}}{2\bar{\sigma}^{2}}\right).
$$
A direct consequence of Proposition~\ref{p:CMG} is the following statement:
\begin{corollary}
Let $\overline{h}$ be the solution of LEG (and RS) filtering problem \eqref{LEGdef} (and \eqref{rde}). Then the following equality holds for any $t \ge 1$:
$$
\overline{h}_t= \widetilde{Z}^{\overline{h}}_{t}.
$$
\end{corollary}
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\section{Particular cases - again}\label{againpart}
\subsection{Markov type observations}
Here we turn to the case when the observations $(Y_t)_{t\ge 1}$ are conditionally independent given $X$. More precisely,
we deal with a signal-observation model
$
(X_t,Y_t)_{t\ge 1},
$
where the signal $X=(X_t)_{t\ge 1},\, X_t \in \mathbb{R}^{n}$ is an arbitrary Gaussian sequence with mean vector
$m=(m_t, t\geq 1)$ and covariance matrix $K =(K(t,s), t\geq 1,
s\geq 1)$, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg X_t=m_t,\quad\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)(X_s-m_s)^{\prime}=K(t,s)\,.
t\geq 1\,,\; s\geq 1\,,
$$
The observation process $Y=(Y_t,\, t\ge 1)$ is given by
\begin{equation}\label{obser}
Y_t= A_t X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t,
\end{equation}
for some sequence $A=(A_t,\, t\ge 1)$ of $m\times n$ matrices, where $\mathop{\mathrm{var}}\nolimitsepsilon=(\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d. $\mathcal{N}(0,Id)$ random variables and $\mathop{\mathrm{var}}\nolimitsepsilon$ and $X$ are independent.
In this case we can write the multidimensional analogue of the equation \eqref{zhtild}, which is nothing else but the dynamic equation for the process $\widetilde{Z}^{h}$ defined by \eqref{Ztildainterpr}. We obtain:
$$
\tilde{Z}^{h}_{t}=m_t +\sum_{l=1}^{t-1}\overline{\gamma}(t,l)[Id+\overline{\gamma}_{l}(A^{\prime}_{l}A_{l}-\mu Q_{l})]^{-1}\mu Q_{l} (h_l - \tilde{Z}_{l}^h) + \sum_{l=1}^{t}\overline{\gamma}(t,l) A^{\prime}_{l} (Y_l - A_l \tilde{Z}_{l}^{h}),
$$
where the matrix $\overline{\gamma}(t,l)$ satisfies the following equation (which is the multidimensional analog of the equation \eqref{GAMMABAR}):
\begin{equation}\label{gammamultdim}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} \overline{\gamma}(t,l)\bar{A}_{l}^{\prime}[Id +\bar{A}_{l}\overline{\gamma}_l\bar{A}_{l}^{\prime}]^{-1}\bar{A}_{l}^{\prime} \overline{\gamma}^{\prime}(s,l),
\end{equation}
where $\bar{A}_{l}=\left(
\begin{array}{c}
A_l \\
-\mu Q_l \\
\end{array}
\right).$
Now the solution of the LEG (and RS) filtering problem $\bar{h}$ is nothing else but:
$$
\bar{h}_{t}=m_t + \sum_{l=1}^{t}\overline{\gamma}(t,l) A^{\prime}_{l} (Y_l - A_l\bar{h}_{l}).
$$
\subsection{Markov type observations, correlated signal and observation noises}
Let us drop the assumption that $X$ and $\mathop{\mathrm{var}}\nolimitsepsilon $ in the observation equation \eqref{obser} are independent. Denote by $K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s)$ the covariance matrix of the signal and the observation noise, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)\mathop{\mathrm{var}}\nolimitsepsilon_s^{\prime}=K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s), \quad
t\geq 1\,,\; s\geq 1.
$$
It can be checked that the following slight modification of the previous statement holds. \\ Let the matrix $\overline{\gamma}(t,l)$ be the unique solution of the following equation
\begin{multline}\label{gammacorrel}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} [\overline{\gamma}(t,l)\bar{A}_{l}^{\prime}+\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)]
\\
[Id +\bar{A}_{l}\overline{\gamma}_l\bar{A}_{l}^{\prime} +\bar{A}_{l}\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)+\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)^{\prime}\bar{A}_{l}^{\prime}]^{-1}
\\
[\bar{A}_{l}^{\prime} \overline{\gamma}^{\prime}(s,l)+\bar{K^{\prime}}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(s,l)],
\end{multline}
with $\bar{A}_{l}=\left(
\begin{array}{c}
A_l \\
-\mu Q_l \\
\end{array}
\right),\quad \bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)=(K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)\quad \mathbf{0}).$\\
Then the solution of the LEG (and RS) filtering problem $\bar{h}$ satisfies the following equation
\begin{equation}\label{hbarcorrel}
\bar{h}_{t}=m_t + \sum_{l=1}^{t}[\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)+\overline{\gamma}(t,l) A^{\prime}_{l}][Id +A_{l}K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)]^{-1} (Y_l - A_l\bar{h}_{l}).
\end{equation}
\subsection{Observations containing Moving Averages of order~1}\label{MA1XY}
Now we consider the case of a MA(1) type process, {\em i.e.}, the following signal-observation model:
$$
X_{t} = \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t} + \lambda \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t-1}\,; t\ge 1\,,
$$
$$
Y_{t} =\alpha_t X_t + \mathop{\mathrm{var}}\nolimitsepsilon_{t} + \beta \mathop{\mathrm{var}}\nolimitsepsilon_{t-1}\,; t\ge 1\,,
$$where $( \mathop{\mathrm{var}}\nolimitsepsilon_t, \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t\ge 0}$~is a sequence of i.i.d.
Gaussian variables and $\lambda$ and $\beta$ are real numbers.\\
Let us denote by $A_t$ the row $\bar{A}_{t}=(\alpha_{t}\quad \beta)$ and by $\bar{X}_{t}$ the vector $\bar{X}_{t}=\left(
\begin{array}{c}
X_t \\
\mathop{\mathrm{var}}\nolimitsepsilon_{t-1} \\
\end{array}
\right).$
Of course
$\bar{X}$ is centered, has the
covariance matrix
$K(t,s) =\left(
\begin{array}{cc}
(1+\lambda^{2}) \mathbf{1}(s=t-1)+ \lambda \mathbf{1}(s=t)& 0 \\
0 & \mathbf{1}(s=t) \\
\end{array}
\right)$
and the covariance between $\bar{X}$ and $\mathop{\mathrm{var}}\nolimitsepsilon$ is $K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s)=\left(
\begin{array}{c}
0 \\
\mathbf{1}(s=t-1) \\
\end{array}
\right).$
The solution $\overline{\gamma}$ to (\ref{gammacorrel}) then can be found as:
$$
\overline{\gamma}(t,s) =\mathbf{0}\,,\; s < t-1\,; \quad \overline{\gamma}(t,t-1) = \left(
\begin{array}{cc}
\lambda & 0 \\
0 & 0 \\
\end{array}
\right)\,,\;
t\ge 1\,,
$$
and $\overline{\gamma}(t,t)=\overline{\gamma}_{t}$ where $\overline{\gamma}_{t}$ is
the solution of the equation:
\begin{multline*}
\overline{\gamma}_{t} =\left(
\begin{array}{cc}
1+\lambda^{2} & 0 \\
0 & 1 \\
\end{array}
\right) +
\\
+\left(
\begin{array}{cc}
\lambda\alpha_{t-1} & -\lambda\mu Q_{t-1}\\
0 & 1 \\
\end{array}
\right)\left[ Id + \left(
\begin{array}{cc}
\alpha_{t-1} & \beta \\
-\mu Q_{t-1} & 0 \\
\end{array}
\right)\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
\alpha_{t-1} & -\mu Q_{t-1} \\
\beta & 0 \\
\end{array}
\right)\right]^{-1}
\\
\times \left(
\begin{array}{cc}
\lambda\alpha_{t-1} & 0 \\
-\lambda\mu Q_{t-1} & 1 \\
\end{array}
\right) \,,\;t\ge 1\,;
\quad \overline{\gamma}_0= \mathbf{0}.
\end{multline*}
provided that this equation has the unique nonnegative definite solution.
Moreover, equation
\eqref{hbarcorrel} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation*}
\overline{h}_t=\Lambda_{t}^{-1} \left(
\begin{array}{c}
\lambda\alpha_{t} \\
1+\beta \\
\end{array}
\right) [Y_{t-1}-A_{t-1}\overline{h}_{t-1}] +\Lambda_{t}^{-1}\overline{\gamma}_t \left(
\begin{array}{c}
\alpha_{t} \\
\beta \\
\end{array}
\right)Y_t, \, t\ge 1,\, \overline{h}_0=0,
\end{equation*}
with $\Lambda_{t}= Id + \overline{\gamma}_{t}A_{t}^{\prime}A_{t}.$
\subsection{Observations containing Gaussian AR(1) process }\label{GMSXY}
In this part we concentrate on the case of a
Gaussian AR(1) type process $Y$, {\em i.e.},
\begin{equation}\label{model ARY}
Y_{t}= \alpha_{t} X_{t}+\mathop{\mathrm{var}}\nolimitsepsilon_{t}\,,\; t\ge 1\,;
\quad Y_{0}=0\,,
\end{equation}
where
$$
\mathop{\mathrm{var}}\nolimitsepsilon_{t}=b \mathop{\mathrm{var}}\nolimitsepsilon_{t-1}+\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t},
$$
and $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables independent of $X$.
We also suppose that the signal $X$ is a Gaussian AR(1) process,
{\em i.e.},
\begin{equation}\label{model ARY}
Y_{t}= a_{t} X_{t}+\epsilon_{t}\,,\; t\ge 1\,;
\quad X_{0}=0\,,
\end{equation}
and also $(\epsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables.
Proceeding as in Sections \ref{GMS} and \ref{MA1XY} we can write the dynamic equation for the solution of LEG and RS filtering problems $\bar{h}$. Namely, $\bar{h}$ is the first component $\bar{h}^{1} $ of the solution of the following recursive equation:
\begin{equation*}
\overline{h}_t=\overline{\gamma}_t A_t^{\prime}[Y_{t}-A_{t}\overline{h}_{t}] +\left(
\begin{array}{cc}
a_{t} & 0 \\
0 & b \\
\end{array}
\right) \overline{h}_{t-1} + \left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right)[Y_{t-1}-A_{t-1}\overline{h}_{t-1}] \, t\ge 1,\, \overline{h}_0=\mathbf{0},
\end{equation*}
where $A_t= (\alpha_{t} \quad \beta)$ and $\overline{\gamma}$ is the unique nonnegative defined solution of the Ricatti equation:
\begin{multline*}
\overline{\gamma}_t=\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right)\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right) -\left[\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right)\overline{\gamma}_{t-1}\bar{A}^{\prime}_{s-1} +\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
\end{array}
\right)\right]
\\
\times\left[Id +\bar{A}_{t-1} \overline{\gamma}_{t-1}\bar{A}^{\prime}_{s-1}\right]^{-1}\left[\bar{A}_{t-1}\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right) +\left(
\begin{array}{cc}
0 & 0 \\
0 & 1 \\
0 & 0 \\
\end{array}
\right)\right],
\end{multline*}
with $\bar{A}_{t}=\left(
\begin{array}{cc}
\alpha_{t} & b \\
-\mu Q_t & 0 \\
0 & 0 \\
\end{array}
\right).$
\end{document}
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\begin{equation}gin{document}
\title{Quantum optical memory protocols in atomic ensembles}
\author{Thierry Chaneli\`ere}
\affiliation{Laboratoire Aim\'e Cotton, CNRS, Univ. Paris-Sud, ENS Paris-Saclay, Universit\'e Paris-Saclay, 91405 Orsay, France}
\author{Gabriel H\'etet}
\affiliation{Laboratoire Pierre Aigrain, Ecole normale sup\'erieure, PSL Research University, CNRS, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'es, Universit\'e Paris Diderot, Sorbonne Paris-Cit\'e, 24 rue Lhomond, 75231 Paris Cedex 05, France}
\author{Nicolas Sangouard}
\affiliation{Quantum Optics Theory Group, Department of Physics, University of Basel, CH-4056 Basel, Switzerland}
\begin{equation}gin{abstract}
We review a series of quantum memory protocols designed to store the {quantum} information carried by {light} into atomic ensembles. {In particular, we show how a simple semiclassical formalism allows to gain insight into various memory protocols and to highlight strong analogies between them. These analogies naturally lead to a classification of light storage protocols in{to} two categories, {namely} photon echo and {\it slow-light} memories.} {We focus on the storage and retrieval dynamics as a key step to map the optical information into the atomic excitation.} We finally {review} various criteria adapted for both continuous variables and photon-counting measurement techniques to {certify} the quantum nature of {these} memory {protocols}.
\end{abstract}
\maketitle
\section{Introduction}
{The potential of quantum information sciences for applied physics is currently highlighted by coordinated and voluntarist policies.} In a global scheme of {probabilistic} quantum information processing, {quantum} memory is a key element {to synchronize independent events \cite{bussieres2013prospective}}. {Memory for light} can be more generally considered as an interface between light (optical or radiofrequency) and a material medium \cite{hammerer2010quantum} where the quantum information is mapped from one form (optical for example) to the other (atomic excitation) and {\it vice versa}. {In this chapter, we review quantum protocols for light storage}. The objective is not to make a comparative and exhaustive review of the different systems {or applications} of interest. Analysis {along these lines} can be found in many review articles \cite{bussieres2013prospective, hammerer2010quantum, lvovsky2009optical, afzelius2010photon, review_Simon2010, review_Heshami_2016, review_Ma_2017} perfectly reflecting the state of the art. Instead, we focus on pioneering protocols {in atomic ensembles} that we analyze with the same formalism to extract the common {features} and differences.
{First,} we consider two representative classes of storage protocols, the photon echo in section \ref{sec:2PE} and the {\it slow-light} memories in section \ref{sec:SL}. In both cases, we first derive a minimalist semi-classical Schr\"odinger-Maxwell model to describe the propagation of a weak signal {in an atomic ensemble}. Two-level atoms are sufficient to characterize the photon echo protocols among which the standard two-pulse photon echo is the historical example (section \ref{sec:2PE}). On the contrary, as in the widely studied {\it stopped-light} by means of electromagnetically induced transparency (EIT), the minimal atomic structure consists of three levels (section \ref{sec:SL}). {In both cases}, however, the semi-classical Schr\"odinger-Maxwell formalism is sufficient to describe the optical storage dynamics and evaluate the theoretical efficiencies.
To fully replace our analysis in the context of the quantum storage, we finally derive a variety of criteria in section \ref{sec:certification} to {certify} the quantum nature of optical memories. Our approach is pragmatic in this section as we do not develop a fully quantized propagation model mirroring our semi-classical analysis in \ref{sec:2PE} and \ref{sec:SL}. Instead, we use an atomic chain quantum toy model to characterize the {noise of various storage protocols}. Criteria depending on experimentally accessible parameters are {reviewed} for both continuous and discrete variables.
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\section{Photon echo memories}\label{sec:2PE}
The photon echo technique as the optical {\it alter ego} of the spin echo has been considered early as a spectroscopic tool \cite{Kopvillem,Hartmann64, Hartmann66, Hartmann68}. Its extensive description can be found in many textbooks as an example of a coherent transient light-atoms interaction \cite{allen2012optical}. Due to its coherent nature and many experimental realizations over the last decades, the photon echo has been reconsidered in the context of quantum storage \cite{afzelius2010photon}. In this section, we will first establish the formalism describing the propagation and the retrieval of week signals in a two-level inhomogeneous atomic medium. We then describe and evaluate the efficiency of the standard two-pulse photon echo from the point of view of a storage protocol. The latter is not immune to noise but has stimulated the design of noise free alternatives, namely the Controlled reversible inhomogeneous broadening and the Revival of silenced echo that we will describe using the same formalism.
The signal propagation and photon echo retrieval can be modeled by the Schr\"odinger-Maxwell equations in one dimension (along $z$) with an inhomogeneously broadened two-level atomic ensemble that we will first illustrate.
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\subsection{Two-level atoms Schr\"odinger-Maxwell model}
On one side, the atomic evolution under the field excitation is given by the Schr\"odinger equation and on the other side, the field propagation is described by the Maxwell equation that we successively remind.
\subsubsection{Schr\"odinger equation for two-level atoms}
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.7\linewidth]{2level_3level.eps}}
\caption{Two-level (left) and three-level atoms (right) used to describe the photon echo (section \ref{sec:2PE}) and the {\it slow-light} memories (section \ref{sec:SL}).The signal $\mathcal{E}$ is applied on the $|g\rangle$ and $|e\rangle$ transition. For the three-level atoms, a control field $\Omega$ is applied on the $|s\rangle$ and $|e\rangle$ transition.}
\label{fig:2level_3level}
\end{figure}
For two-level atoms, labeled $|g\rangle$ and $|e\rangle$ for the ground and excited states (see fig.\ref{fig:2level_3level}, left), the rotating-wave probability amplitudes $C_g$ and $C_e$ respectively are governed by the time-dependent Schr\"odinger equation \cite[eq. (8.8)]{shore2011manipulating}:
\begin{equation}gin{align}
i \partial_t \left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
0 &\displaystyle \frac{\mathcal{E}^*}{2} \\
\displaystyle\frac{\mathcal{E}}{2} & -\Delta \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
\end{array}\right]
\label{bloch2}\end{align}
where $\mathcal{E}(z,t)$ is the complex envelope of the input signal expressed in units of Rabi frequency. $\Delta$ is the laser detuning.
The atomic variables $C_g$ and $C_e$ depend on $z$ and $t$ for a given detuning $\Delta$. The detunings can be made time-dependent
\cite{loy1974observation,vitanov2001laser}, position-dependent or both \cite{hetet2008electro} but this is not the case here.
Decay terms can be added {by-hand} by introducing a complex detuning $\Delta \rightarrow \Delta - i \Gamma$ where $\Gamma$ is the decay rate of the excited state $|e\rangle$ \footnote{We do not distinguish the decay terms for the population and the coherence. This is an intrinsic limitation of the Schr\"odinger model as opposed to the density matrix formalism (optical Bloch equations).}.
\subsubsection{Maxwell propagation equation}
The propagation of the signal $\mathcal{E}(z,t)$ is described by the Maxwell equation that can be simplified in the slowly varying envelope approximation \cite[eq. (21.15)]{shore2011manipulating}. This reads for an homogeneous ensemble whose linewidth is given by the decay term $\Gamma$:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle {i \alpha} \Gamma C_g^* C_e \label{MB_M_hom}
\end{equation}
The term $C_g C_e^* $ is the atomic coherence on the $|g\rangle \rightarrow |e\rangle$ transition directly proportional to the atomic polarization. The light coupling constant is included in the absorption coefficient $\alpha$ (inverse of a length unit), thus the right hand side represents the macroscopic atomic polarization.
The Maxwell equation can be generalized to an inhomogeneously broaden ensemble \cite{allen2012optical}:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) C_g^* C_e \mathrm{d}\Delta \label{MB_M_inhom}
\end{equation}
where $ g\left(\Delta\right) $ is the normalized inhomogeneous distribution.
Photon echo memories precisely rely on the inhomogeneous broadening as an incoming bandwidth. The set of equations (\ref{bloch2}\&\ref{MB_M_inhom}) are then relevant in that case. The resolution can be further simplified for weak $\mathcal{E}(z,t)$ signals as expected for quantum storage. This is the so-called perturbative regime. More importantly, the perturbative limit is necessary to ensure the linearity of the storage scheme and is then not only a formal simplification. The perturbative expansion should be used with precaution when photon echo protocols are considered. When strong (non-perturbative) $\pi$-pulses are used to trigger the retrieval as a coherence rephasing, they unavoidably invert the population. This interplay between rephasing and inversion is the essence of the photon echo technique. Population inversion should be avoided because spontaneous emission induces noise \cite{ruggiero}. We will nevertheless first consider the standard two-pulse photon echo scheme because this is the ancestor and an inspiring source for modified photon echo schemes adapted for quantum {storage}.
\subsubsection{Coherent transient propagation in an inverted or non-inverted medium}
The goal of the present section is to describe the propagation of a weak signal representing both the incoming signal and the echo. For the standard two-pulse photon echo (see section \ref{2PE}), the echo is emitted in an inverted medium so we will consider both an inverted and a non-inverted medium corresponding to the ideal storage scheme (see \ref{CRIB} and \ref{ROSE}).
The propagation is coherent in the sense that the pulse duration is much shorter than the coherence time. The decay term (that could be introduced with a complex detuning $\Delta$) is fully neglected in eq.\eqref{bloch2}.
The coherent propagation is defined in the perturbative regime. This latter should be defined with precaution if the medium is inverted or not. The coherence term $\mathcal{P}=C_g^* C_e$ appearing in the propagation equation (eq.\ref{MB_M_hom} or \ref{MB_M_inhom}) is described by rewriting the Schr\"odinger equation
as
\begin{equation}gin{equation}
\partial_t \mathcal{P} =i\Delta \mathcal{P} + \left( C_e^* C_e-C_g^* C_g \right) i \frac{\mathcal{E}}{2}
\end{equation}
The reader more familiar with the optical Bloch equations can directly recognize the evolution of the coherence term (non-diagonal element of the density matrix) where the term $ \left( C_e^* C_e-C_g^* C_g \right) $ is the population difference (diagonal element{s}).
For a non-inverted medium, the atoms are essentially in the ground state, so in the perturbative limit $ \left( C_e^* C_e-C_g^* C_g \right) \rightarrow -1$. The population goes as the second order in field excitation thus justifying the perturbative expansion where the coherences $\mathcal{P}$ goes as the first order. Along the same line, $ \left( C_e^* C_e-C_g^* C_g \right) \rightarrow 1$ for an inverted medium. The atomic evolution reads as
\begin{equation}gin{equation}
\partial_t \mathcal{P} =i\Delta \mathcal{P} \mp i \frac{\mathcal{E}}{2}
\end{equation}
where $\mp$ indicates if the medium is non-inverted (ground state) or inverted (excited state). This can be alternatively written in an integral form as
\begin{equation}gin{equation}
\mathcal{P}(z,t) =\mp \frac{i}{2} \int_{-\infty}^t \mathcal{E}\left(z,t^\prime\right) \exp \left(i\Delta\left(t-t^\prime\right)\right) \ensuremath{\mathrm{d}} t^\prime \label{integral_form}
\end{equation}
As given by eq.\eqref{MB_M_inhom}, the propagation in the inhomogeneous medium is described by
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}_\Delta(z,t) \ensuremath{\mathrm{d}} \Delta \label{MB_M_inhom_P}
\end{equation}
We remind by an index $ \mathcal{P}_\Delta$ that the coherence term depends on the detuning $\Delta$ as a parameter.
To avoid the signal temporal distortion, the incoming pulse bandwidth should be narrower than the inhomogeneous broadening given by the distribution $g\left(\Delta\right) $ so we can safely assume $g\left(\Delta\right) \rightarrow 1$. The double integral term $\displaystyle \int_\Delta \mathcal{P}_\Delta \ensuremath{\mathrm{d}} \Delta$ from eq.\eqref{integral_form} can be simplified by writing $\displaystyle \int_\Delta \exp \left(i\Delta\left(t-t^\prime\right)\right) \ensuremath{\mathrm{d}} \Delta \rightarrow 2 \pi \delta_{t^\prime=t} $ as a representation of the Dirac peak $\delta_0$
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
\mp\displaystyle\frac{\alpha}{2} \mathcal{E}(z,t)\label{bouguer0}
\end{equation}
Eq.\eqref{bouguer0} is the absorption law or gain if the medium is inverted. The absorption law was at first discovered by Bouguer \cite{bouguer1760traite}, today known as the Bouguer-Beer-Lambert law. The description can be even more simplified by noting that the pulse length is usually much longer the medium spatial extension. The term $\displaystyle \frac{1}{c}\partial_t$ can be dropped leading to the canonical version of the Bouguer-Beer-Lambert law \cite{allen2012optical}.
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)=
\mp\displaystyle\frac{\alpha}{2} \mathcal{E}(z,t)\label{bouguer}
\end{equation}
This form can be alternatively obtained by writing the equation in the moving frame at the speed of light. Introducing the moving frame may be a source of mistake when the backward retrieval configuration is considered (see section \ref{CRIB}). {Anyway,} the moving frame {does not need to be introduced} because the medium length $L$ is in practice much shorter than the pulse extension. In other words, the delay induced by the propagation $L/c$ is negligible with respect to the pulse duration. The term {\it propagation} is in that case arguable when the term $\displaystyle \frac{1}{c}\partial_t$ is absent. Propagation should be considered in the general sense. The absorption coefficient in eq.\eqref{bouguer} defines a propagation constant. This latter is real as opposed to a propagation delay which would appear as a complex (purely imaginary) constant.
The Bouguer-Beer-Lambert law can be obtained equivalently with an homogeneous medium including the coherence decay term. This is not the case here. We insist: there is no decay and the evolution is fully coherent. To illustrate this fundamental aspect of the coherent propagation, we can show that the field excitation is actually recorded into the medium. On the contrary, with a decoherence term, the field excitation would be lost in the environment. The complete field excitation to coherence mapping is a key ingredient of the photon echo memory scheme.
\subsubsection{Field excitation to coherence mapping}\label{mapping}
In the coherent propagation regime, the evolution of the atomic and optical variables is fully coherent. Let {us} restrict the discussion to the case of interest, namely the photon echo scheme of an initially non-inverted (ground state) medium. The field is absorbed following the Bouguer-Beer-Lambert law (eq.\ref{bouguer}). This disappearance of the field is not due to the atomic dissipative decay but to the inhomogeneous dephasing. For example, in an homogeneous sample, the absorption of the laser beam can be due to spontaneous emission: the beam is depleted because the photons are scattered in other modes. In an inhomogeneous sample, the beam depletion is due to dephasing and not dissipation. In other words, the forward scattered dipole emissions destructively interfere. Since the evolution is coherent, the field should be fully mapped into the atomic excitation. In that case, the expression \eqref{integral_form} can be reconsidered by noting that after the absorption process, the integral boundary can be pushed to $+\infty$ as
\begin{equation}gin{align}
\mathcal{P}_\Delta(z,t)& =- \frac{i}{2} \exp \left(i \Delta t \right) \int_{-\infty}^{+\infty} \mathcal{E}\left(z,t^\prime\right) \exp \left(- i\Delta t^\prime \right) \ensuremath{\mathrm{d}} t^\prime \\ &=- \frac{i}{2} \exp \left(i \Delta t \right) \tilde{\mathcal{E}}(z,\Delta) \label{mapping}
\end{align}
where $ \tilde{\mathcal{E}}(z,\omega)$ is the Fourier transform of the incoming pulse $\displaystyle \mathcal{E}\left(z,t^\prime\right)$ \footnote{ We define the Fourier transform pairs as \begin{equation}gin{align}
\tilde{f}(\omega)&=\int_t f(t) \exp( -i\omega t)\mathrm{d} t\\
{f}(t)&=\frac{1}{2\pi}\int_\omega \tilde{f}(\omega) \exp(i\omega t)\mathrm{d} \omega
\end{align}
}. This expression tells that the incoming spectrum is entirely mapped into the atomic excitation. More precisely, each class $\mathcal{P}_\Delta$ in the atomic distribution actually records the corresponding part in the incoming spectrum $\tilde{\mathcal{E}}(z,\Delta)$. The term $\exp \left(i \Delta t \right)$ simply reminds us that the coherence freely oscillates after the field excitation. An exponential decay term could be added by-hand by giving an imaginary part to the detuning $\Delta$.
This mapping stage when the field is recorded into the atomic coherences of an inhomogeneous medium is the initial step of the different photon echo memory schemes. Various techniques have been developed to retrieve the signal after the initial absorption stage. The inhomogeneous dephasing is the essence of the field to coherence mapping since the field spectrum is recorded in the inhomogeneous distribution. The retrieval is in that sense always associated to a rephasing or compensation of the inhomogeneous dephasing. This justifies the term photon echo used to classify this family of protocols. We will start by describing the standard two-pulse photon echo (2PE). Despite a clear limitation for quantum storage, this is an enlightening historical example. Its descendants as the so-called Controlled reversible inhomogeneous broadening (CRIB) and Revival of silenced echo (ROSE) have been precisely designed to avoid the deleterious effect of the $\pi$-pulse rephasing used in the 2PE sequence.
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\subsection{Standard two-pulse photon echo}\label{2PE}
Inherited from the magnetic resonance technique \cite{hahn}, the coherence rephasing {and the subsequent field reemission} is triggered by applying a strong $\pi$-pulse (fig.\ref{fig:2PE}). The possibility to use the 2PE for pulse storage has been mentioned early in the context of optical processing \cite{Carlson:83}. The retrieval efficiency can indeed be remarkably high \cite{moiseev1987some,azadeh, SjaardaCornish:00}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{2PE.eps}}
\caption{Standard two-pulse photon echo sequence. Inherited from magnetic resonance, $\pi/2-\pi$ sequence (delayed by $\tau$, in magenta) produces an echo at $2\tau$ (in red). When considered for optical storage, the first pulse is weak (in blue) and longer than the rephasing $\pi$-pulse.}
\label{fig:2PE}
\end{figure}
This particularity has attracted a renewed curiosity in the context of quantum information \cite{rostovtsev2002photon, moiseev_echo_03}.
\subsubsection{Retrieval efficiency}
The retrieval efficiency can be derived analytically from the Schr\"odinger-Maxwell model. Following the sequence in fig.\ref{fig:2PE}, the signal absorption is first described by the Bouguer-Beer-Lambert law (eq.\ref{bouguer}). The initial stage is followed by a free evolution during a delay $\tau$. The $\pi$-pulse will trigger a retrieval. The action of a strong pulse on the atomic variables is described by the propagator
\begin{equation}gin{align}
\left[
\begin{equation}gin{array}{c}
C_g\left(\tau^+\right) \\
C_e\left(\tau^+\right) \\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
\displaystyle \cos(\theta/2) & \displaystyle -i \sin(\theta/2) \\
\displaystyle -i \sin(\theta/2) &\displaystyle \cos(\theta/2) \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \left(\tau^-\right) \\
C_e \left(\tau^-\right) \\
\end{array}\right]
\end{align}
which links the atomic variables just before ($\tau^-$) and after ($\tau^+$) a general $\theta$-area pulse. This solution of the canonical Rabi problem is only valid for a {very short} pulse (hard pulse). More precisely, in the atomic evolution eq.\eqref{bloch2}, the Rabi frequency {must be} much larger than the detuning. In the 2PE scheme, this means that the atoms excited by the signal (first pulse) are uniformly (spectrally) covered by the strong rephasing pulse. This translates in the time domain as a condition on the relative pulse durations: the $\pi$-pulse {must} be much shorter than signal. This aspect appears as an initial condition for the pulse durations but is also intimately related to the transient coherent propagation of strong pulses among which $\pi$-pulses are a particular case. This will be discussed in more details in the appendix \ref{strong_pulse}. Assuming the ideal situation of the uniform $\theta=\pi$ pulse area, the propagator takes the simple form
$$
\left[
\begin{equation}gin{array}{ccc}
0 & -i \\
-i & 0 \\
\end{array}\right]$$
fully defining the effect of the $\pi$-pulse on the stored coherence
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^+\right)=-\mathcal{P}_\Delta^*\left(\tau^-\right) =- \frac{i}{2} \exp \left(-i \Delta \tau \right) \tilde{\mathcal{E}}^*(z,\Delta)\label{P_tauplus}
\end{equation}
{T}he free evolution resumes by adding the inhomogeneous phase $\Delta \left( t- \tau\right)$
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau\right)=\mathcal{P}_\Delta\left(\tau^+\right) \exp \left(i \Delta \left( t- \tau\right) \right) =- \frac{i}{2} \exp \left(i \Delta \left( t- 2\tau\right) \right) \tilde{\mathcal{E}}^*(z,\Delta)\label{P_echo}
\end{equation}
In the expression \eqref{P_echo}, we see that the inhomogeneous phase $\Delta \left( t- 2\tau\right)$ is zero at the instant $t=2\tau$ of the retrieval thus justifying the term rephasing.
The propagation of the retrieved echo $\mathcal{E}^R$ follows eq.\eqref{MB_M_inhom_P}. The source term on the right-hand side has now two contributions. The first one gives the Bouguer-Beer-Lambert law (eq \ref{bouguer}) for the echo field $\mathcal{E}^R$ itself. A critical aspect of the 2PE is the population inversion induced by the $\pi$-pulse. The intuition can be confirmed by calculating from the propagator $ \left( C_e^*\left(\tau^+\right) C_e\left(\tau^+\right)-C_g^*\left(\tau^+\right) C_g\left(\tau^+\right) \right)$ to the first order by noting that $ C_g\left(\tau^-\right) \simeq 1$. The echo field $\mathcal{E}^R$ exhibits gain. The second one comes from the coherence initially excited by the signal freely oscillating after the $\pi$-pulse rephasing. In other words, the coherences at the instant of retrieval are the sum of the free running term due to the signal excitation from eq.\eqref{P_echo} and the contribution from the echo field itself.
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
+\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) -\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}_\Delta(z,t>\tau) \ensuremath{\mathrm{d}} \Delta
\end{equation}
The integral source term representing the build-up of the macroscopic polarisation at the instant of retrieval is directly related to the signal field excitation $\mathcal{E}$ which appears as the inverse Fourier transform of $\tilde{\mathcal{E}}^*(z,\Delta)$ from eq.\eqref{P_echo}, {that is}
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
+\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}^*(z,2\tau-t) \label{eq_echo}
\end{equation}
{Eq.\eqref{eq_echo} is simple but rich because it can be modified by-hand to describe the descendants of the 2PE protocol that are suitable for quantum storage as we will see in sections \ref{CRIB} and \ref{Rose}. Note that it can be adapted to account for rephasing pulse areas $\theta$ that are not $\pi.$ They lead to imperfect rephasing and incomplete medium inversion thus modifying the terms in eq.\eqref{eq_echo} \cite{ruggiero}. Very general expressions for the efficiency as a function of $\theta$ can be analytically derived \cite{moiseev1987some}.}
{Knowing that the incoming signal follows the Bouguer-Beer-Lambert law (eq.\ref{bouguer}) of absorption $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right),$ the efficiency of the 2PE can be obtained as a function of optical depth $d=\alpha L$ from the ratio between the output and input intensities
\begin{equation}gin{equation}
\eta=\frac{|\mathcal{E}^R(L,t)|^2}{|\mathcal{E}(0,2\tau-t)|^2}\label{efficiency}
\end{equation}
For a $\pi$-rephasing pulse, we find
\begin{equation}gin{equation}\label{etaPi}
\eta\left(d\right)=\left[\exp\left(d/2\right)-\exp\left(-d/2\right)\right]^2 =4~{\rm sinh}^2\left(d/2\right)
\end{equation}
}
{At large optical depth $d$, the efficiency scales as $\exp\left(d\right)$ resulting in an exponential amplification of the input field. This amplification prevents the 2PE to be used as a quantum storage protocol. The simplest but convincing argument uses the no-cloning theorem \cite{nocloning}. Alternatively, we can apply various criteria to certify the quantum nature of the memory on the echo and show that none of these criteria witnesses its non-classical feature, as wee will see section \ref{sec:certification}.}
In fig.\ref{fig:2PE_simul} (bottom), we have represented this efficiency scaling (eq.\ref{etaPi}) that we compare with a numerical simulation of a 2PE sequence solving the Schr\"odinger-Maxwell model. For a given inhomogeneous detuning $\Delta$, we calculate the atomic evolution eq.\eqref{bloch2} by using a fourth-order Runge-Kutta method. After summing over the inhomogeneous broadening, the output pulse is obtained by integrating eq.\eqref{MB_M_inhom} along $z$ using the Euler method.
In the numerical simulation, there is no assumption on the $\pi$-pulse duration with respect to the signal bandwidth (as needed to derive the analytical formula eq.\ref{etaPi}). The excitation pulses are assumed Gaussian as shown for the incoming and the outgoing pulses of a 2PE sequence after propagation though an optical depth $d= 2$ (fig.\ref{fig:2PE_simul}, top). We consider different durations for the $\pi$-pulse (of constant area) and a fixed signal duration.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.85\linewidth]{2PE_simul.eps}}
\caption{Top: Numerical simulation of a 2PE sequence with a weak incoming signal (area $\pi/20$). The signal and the echo fields are in blue and have been magnified by a factor 10 (shaded area). The incoming pulses are in solid lines. The outgoing pulses after propagation though $d = 2$ are in dashed line. The $\pi$-pulse is two times shorter than the signal. Bottom: Storage efficiencies (see text for the definition) as a function of the optical depth $d$ . The black line is the analytical solution eq.\eqref{etaPi}. Three simulations have been performed depending on the relative duration of the $\pi$-pulse with respect to the signal: when the $\pi$-pulse has the same duration than the signal (ratio 1), when it is 2 times (ratio 2) and 10 times shorter (ratio 10). The circle corresponds to the 2PE sequence on top.}
\label{fig:2PE_simul}
\end{figure}
From the numerical simulation, the efficiency is evaluated by integrating under the intensity curves of the echo (shades area). This latter reaches 152\% for the sequence of fig.\ref{fig:2PE_simul} (top), larger than 100\% as expected for an inverted medium. Still, this is much smaller than the 552\% {efficiency} expected from eq.\eqref{etaPi} with $d= 2$. This discrepancy is essentially explained by the $\pi$-pulse distortion through propagation (magenta dashed line in fig.\ref{fig:2PE_simul},top) than can be observed numerically. The $\pi$-pulse should stay shorter than the signal to properly ensure the coherence rephasing. This is obviously not the case because the pulse is distorted as we briefly analyze in appendix \ref{strong_pulse} with the energy and area conservation laws.
As a summary, we have evaluated numerically the efficiencies when the $\pi$-pulse has the same duration than the signal (ratio 1), when it is 2 times and 10 times shorter (ratio 2 and 10 respectively). We see in fig.\ref{fig:2PE_simul} (bottom) than the efficiencies deviates significantly from the prediction eq.\eqref{etaPi}. There is less discrepancy when the $\pi$-pulse is 10 times shorter than the signal (ratio 10), especially at low optical depth. Still, for larger $d$, the distortions are sufficiently important to reduce the efficiency significantly.
Despite a clear deviation from the analytical scaling (eq.\ref{etaPi}), the echo amplification is important (efficiency $>$ 100\%). This latter comes from the inversion of the medium. As a consequence, the amplified spontaneous emission mixes up with the retrieved signal then inducing noise. It should be noted that the signal to noise ratio only depends on the optical depth \cite{ruggiero,RASE,Sekatski}. This may be surprising at first sight because the coherent emission of the echo and the spontaneous emission seems to have completely different collection patterns offering a significant margin to the experimentalist to filter out the noise. This is not the case. The excitation volume is defined by the incoming laser focus. On the one hand, a tighter focus leads to a smaller number of inverted atoms thus reducing the number spontaneously emitted photons. On the other hand, a tight focus requires a larger collection angle of the retrieved echo. Less atoms are excited but the spontaneous emission collection angle is larger. The noise in the echo mode is unchanged. This qualitative argument which can be seen as a conservation of the optical etendue is quantitatively supported by a quantized version of the Bloch-Maxwell equations \cite{RASE,Sekatski}. This aspect will be discussed in sections \ref{CV_criterion} and \ref{counting_criterion} using a simplified quantum model.
In any case, population inversion should be avoided. This statement motivated many groups to conceive rephasing protocols by keeping the best of the 2PE but avoiding the deleterious effect of $\pi$-pulses as we will see now in sections \ref{CRIB} and \ref{Rose}.
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\subsection{Controlled reversible inhomogeneous broadening}\label{CRIB}
The controlled reversible inhomogeneous broadening (CRIB) offers a solid alternative to the 2PE \cite{CRIB1,CRIB2,CRIB3,CRIB4, sangouard_crib}. The CRIB,,as represented in fig.\ref{fig:CRIB}, has been successfully implemented with large efficiencies \cite{hedges2010efficient, hosseini2011high} and low noise measurements \cite{lauritzen2010telecommunication} validating the protocol as a quantum memory in different systems, from atomic vapors to doped solids \cite{lauritzen2011approaches}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{CRIB.eps}}
\caption{CRIB echo sequence. As compared to the 2PE sequence, no rephasing pulse is applied, the inhomogeneous broadening is reversed using a controllable electric field for example \cite{CRIB3} (magenta line).}
\label{fig:CRIB}
\end{figure}
Fundamentally, an echo is generated by rephasing the coherences corresponding to the cancellation of the inhomogeneous phase. As indicated by eq.\eqref{mapping}, the accumulated phase is $\Delta t$. Taking control of the detuning $\Delta$ is sufficient to produce an echo without a $\pi$-pulse. This is the essence of the CRIB sequence, where the detuning is actively switched from $\Delta$ for $t<\tau$ to $-\Delta$ for $t>\tau$. We won't focus on the realization of the detuning inversion. This aspect has been covered already and we recommend the reading of the review papers \cite{lvovsky2009optical, afzelius2010photon}. We here focus on the coherence rephasing and evaluate the efficiency which can be compared to other protocols. It should be noted that the gradient echo memory scheme (GEM) \cite{hetet2008electro} is not covered by our description. We will assume that the coherences undergo the transform $+\Delta \rightarrow -\Delta$ independently of the atomic position $z$. This is not the case for the GEM where the detuning $\Delta$ goes linearly (or at least monotonically) with the position $z$. The GEM can be called the longitudinal CRIB. This specificity of the GEM makes it remarkably efficient \cite{hetet2008electro, hedges2010efficient, hosseini2011high}.
Assuming that $\Delta \rightarrow -\Delta$ for $t>\tau$, it should be first noted that at the switching time $\tau$, the coherence term is continuous
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^+\right)=\mathcal{P}_\Delta\left(\tau^-\right) =- \frac{i}{2} \exp \left(i \Delta \tau \right) \tilde{\mathcal{E}}(z,\Delta)
\end{equation}
but will evolve with a different detuning afterward, that is
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau\right)=\mathcal{P}_\Delta\left(\tau^+\right) \exp \left(-i \Delta \left( t- \tau\right) \right) =- \frac{i}{2} \exp \left(i \Delta \left(2\tau -t\right) \right) \tilde{\mathcal{E}}(z,\Delta)\label{P_crib}
\end{equation}
The latter gives the source term of the differential equation defining the efficiency similar to eq.\eqref{eq_echo} for the 2PE
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
-\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}(z,2\tau-t) \label{eq_crib}
\end{equation}
Eqs \eqref{eq_echo} and \eqref{eq_crib} are very similar. The first term on the right hand side is now negative (proportional to $-\displaystyle\frac{\alpha}{2}$) because the medium is not inverted in the CRIB sequence. This is a major difference. Again, the incoming signal follows the Bouguer-Beer-Lambert law of absorption $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right)$ but the efficiency defined by \eqref{efficiency} is now {given} after integration by
\begin{equation}gin{equation}\label{eta_crib}
\eta\left(d\right)=d^2 \exp\left(-d\right)
\end{equation}
The maximum efficiency is obtained for $d=\alpha L=2$ with $\eta\left(2\right)=54\%$ \cite{sangouard_crib} (see fig.\ref{fig:compar_eff}). There is no gain so the semi-classical efficiency is always smaller than one. The efficiency is limited in the so-called forward configuration because the echo is {\it de facto} emitted in an absorbing medium. The re-absorption of the echo limits the efficiency to $54\%$. Ideal echo emission with unit efficiency can be obtained in the backward configuration. This latter is implemented by applying auxiliary pulses, typically Raman pulses modifying the phase matching condition from forward to backward echo emission. The Raman pulses increase the storage time by shelving the excitation into nuclear spin state for example. This ensures the complete reversibility by flipping the apparent temporal evolution (as shown by eq.\eqref{P_crib}) and the wave-vector \cite{reversibility}.
Despite its simplicity, eq.\eqref{eq_crib} can be adapted to describe the backward emission without working out the exact phase matching condition. We consider the following equivalent situation. The signal is first absorbed: $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right)$. We now fictitiously flip the atomic medium: the incoming slice $z=0$ becomes $z=L$ and {\it vice versa}. The atomic excitation would correspond to the absorption of a backward propagating field
$$\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(\alpha \left(z-L\right)/2\right)$$
Eq.\eqref{eq_crib} can be integrated with this new boundary condition, giving the backward efficiency of the CRIB
\begin{equation}gin{equation}\label{eta_crib_back}
\eta\left(d\right)=\left[1 - \exp\left(-d\right)\right]^2
\end{equation}
For a sufficiently large optical depth, the efficiency is close to unity. As a comparison, we have represented the forward (eq.\ref{eta_crib}) and backward (eq.\ref{eta_crib_back}) CRIB efficiencies in fig.\ref{fig:compar_eff}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{compar_eff.eps}}
\caption{Comparison of the forward (eq.\ref{eta_crib}, in blue) and backward (eq.\ref{eta_crib_back}, in red) CRIB efficiency scaling. The standard 2PE efficiency is represented as a reference (eq.\ref{etaPi}, in black)}
\label{fig:compar_eff}
\end{figure}
The practical implementation of the CRIB requires to control dynamically the detuning by Stark or Zeeman effects. The {\it natural} inhomogeneous broadening has a static microscopic origin and cannot be used {\it as it is}. The initial optical depth has to be sacrificed to obtain an effective controllable broadening. This statement motivates the reconsideration of the 2PE which precisely exploit the bare inhomogeneous broadening offering advantages in terms of available optical depth and bandwidth.
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\subsection{Revival of silenced echo}\label{ROSE} \label{Rose}
The Revival of silenced echo (ROSE) is a direct descendant of the 2PE \cite{rose}. The ROSE is essentially a concatenation of two 2PE sequences as represented in fig.\ref{fig:ROSE}. In practice, the ROSE sequence advantageously replace{s} $\pi$-pulses by complex hyperbolic secant (CHS) pulses as we will specifically discuss in \ref{strong_pulse_rose}. For the moment, we assume that the rephasing pulses are simply $\pi$-pulses. This is sufficient to evaluate the efficiency and derive the phase matching conditions.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.85\linewidth]{ROSE.eps}}
\caption{A schematic ROSE echo sequence that can be seen as the concatenation of two 2PE sequences (fig.\ref{fig:2PE}). The first echo at $t=\tau$ (in dashed red) should be silenced by the phase matching conditions (see \ref{phase_match}). A second $\pi$-pulse at $t=\tau^\prime$ induces the emission of the ROSE echo at $t=2\left(\tau^\prime-\tau\right)$ (in red).}
\label{fig:ROSE}
\end{figure}
Concatenated with a 2PE sequence, a second $\pi$-pulse (at $t=\tau^\prime$ in fig.\ref{fig:ROSE}) triggers a second rephasing of the coherences at $t=2\left(\tau^\prime-\tau\right)$. This latter leaves the medium non-inverted avoiding the deleterious effect of a single 2PE sequence. This reasoning is only valid if the first echo is not emitted. In that case, the coherent free evolution continues after the first rephasing. The first echo is said to be silent (giving the name to the protocol) because the coherence rephasing is not associated to a field emission. The phase matching conditions are indeed designed to make the first echo silent but preserve the final retrieval of the signal. Along the same line with the same motivation, McAuslan {\it et al.} proposed to use the Stark effect to silence the emission of the first echo \cite{HYPER} by cunningly applying the tools developed for the CRIB to the 2PE, namely by inducing an artificial inhomogeneous reversible broadening. The AC-Stark shift (light shift) also naturally appeared as a versatile tool to manipulate the retrieval \cite{Chaneliere:15}. We will discuss the phase matching conditions latter. Before that, we will evaluate the retrieval efficiency applying the method developed for the 2PE and CRIB.
\subsubsection{Retrieval efficiency}
Following the procedure in section \ref{2PE}, we assume that a second $\pi$-pulse is applied at $t=\tau^\prime$. Starting from eq.\eqref{P_echo}, we can track the inhomogeneous phase at $t=\tau^\prime$ when the $\pi$-pulse is applied (similar to eq.\ref{P_tauplus}) as
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^{\prime+}\right)=-\mathcal{P}_\Delta^*\left(\tau^{\prime-}\right) =- \frac{i}{2} \exp \left(-i \Delta \left( \tau^\prime -2\tau\right) \right) \tilde{\mathcal{E}}^*(z,\Delta)
\end{equation}
freely evolving afterward as
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau^\prime\right)=- \frac{i}{2} \exp \left(i \Delta \left( t- 2\tau^\prime+2\tau\right) \right) \tilde{\mathcal{E}}(z,\Delta)\label{P_rose}
\end{equation}
There is indeed a rephasing at $ t=2\left(\tau^\prime-\tau\right)$. The retrieval follows the common differential equation (as eqs. \eqref{eq_echo} and \eqref{eq_crib})
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
-\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}(z,t- 2\tau^\prime+2\tau) \label{eq_rose}
\end{equation}
As compared to the 2PE, the ROSE echo is not emitted in an inverted medium. One can note that the signal is not time-reversed as in the 2PE and CRIB, so the efficiency is defined as
\begin{equation}gin{equation}
\eta=\frac{|\mathcal{E}^R(L,t)|^2}{|\mathcal{E}(0,t- 2\tau^\prime+2\tau)|^2}
\end{equation}
The ROSE efficiency is exactly similar to CRIB due to the similarity of eqs.\eqref{eq_crib} and \eqref{eq_rose}. It is limited to 54\% in the forward direction because the medium is absorbing. {Complete reversal can be obtained in the backward direction by precisely designing the phase matching condition, the latter being a critical ingredient of the ROSE protocol.}
Even if there is no population inversion at the retrieval, the use of strong pulses for the rephasing is a potential source of noise. First of all, any imperfection of the $\pi$-pulses may leave some population in the excited state leading to a partial amplification of the signal. Secondarily, the interlacing of strong and weak pulses within the same temporal sequence is like playing with fire. This is a common feature of many quantum memory protocols for which control fields may leak in the signal mode. Many experimental techniques are combined to isolate the weak signal: different polarization, angled beams (spatial selection) and temporal separation. Encouraging demonstrations of the ROSE down to few photons per pulses have been performed by combining theses techniques \cite{bonarota_few}, thus showing the {potentials} of the protocol.
\subsubsection{Phase matching conditions}\label{phase_match}
Phase matching can be considered in a simple manner by exploiting the spectro-spatial analogy. Each atom in the inhomogeneous medium is defined by its detuning (frequency) and position (space), both contributing to the inhomogeneous phase. In that sense, the instant of emission can be seen as a spectral phase matching condition. Following this analogy, the spatial phase matching condition can be derived from the photon echo time sequence \cite{mukamel}.
Let {us} take the 2PE as an example (fig.\ref{fig:2PE}). The 2PE echo is emitted at $t=t_1+2\tau=2t_2-t_1$ where $t_1$ is the arrival time of the signal (first pulse) and $t_2$ the $\pi$-pulse (second pulse). In fig.\ref{fig:2PE}, we have chosen $t_1=0$ and $\tau=t_2-t_1$ for simplicity . By analogy, the echo should be emitted in the direction $\overrightarrow{k}=2\overrightarrow{k_2}-\overrightarrow{k_1}$ where $\overrightarrow{k_1}$ and $\overrightarrow{k_2}$ are the wavevectors of the signal and $\pi$-pulse respectively. In that case, if $\overrightarrow{k_1}$ and $\overrightarrow{k_2}$ are not collinear ($\overrightarrow{k_1}\neq \overrightarrow{k_2}$), the phase matching cannot be fulfilled: there is no 2PE echo emission.
Following the same procedure, the ROSE echo is emitted at $t=t_1+2(\tau^\prime-\tau)=t_1+2(t_3-t_2)$ where $t_3$ is the arrival time of the second $\pi$-pulse (third pulse). The ROSE echo should be emitted if the $\overrightarrow{k}=\overrightarrow{k_1}+2(\overrightarrow{k_3}-\overrightarrow{k_2})$ direction ($\overrightarrow{k_3}$ is the direction of the second $\pi$-pulse). The canonical experimental situation satisfying the ROSE phase matching condition corresponds to $\overrightarrow{k_1}\neq \overrightarrow{k_2}$ (not collinear) but keeping $\overrightarrow{k_3}=\overrightarrow{k_2}$ \cite{Dajczgewand:14, Gerasimov2017}. There is no 2PE in that case because $\overrightarrow{k_1}\neq \overrightarrow{k_2}$ but the ROSE echo is emitted in the direction $\overrightarrow{k_1}$ of the signal as represented in fig.\ref{fig:phase_matching}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{phase_matching.eps}}
\caption{ROSE Phase matching conditions. Left: canonical experimental situation where the two $\pi$-pulses are on the same beam. The echo $\vec{k}$ is in the signal mode $\vec{k_1}$ (forward). The angle between the signal and the rephasing can be large for a good isolation of the echo as recently tested in the orthogonal configuration \cite{Gerasimov2017}. Right: backward retrieval of the ROSE echo $\vec{k}$. The signal $\vec{k_1}$ and two rephasing beams forms a equilateral triangle in that case: the echo is emitted backward.}
\label{fig:phase_matching}
\end{figure}
The backward retrieval configuration is illustrated as well in fig.\ref{fig:phase_matching} (right). The efficiency can reach 100\% because the reversibility of the process is ensured spatially and temporally.
\subsubsection{Adiabatic pulses}\label{strong_pulse_rose}
Even if the protocol can be understood with $\pi$-pulses, the rephasing pulses can be advantageously replaced by complex hyperbolic secant (CHS) in practice \cite{Dajczgewand:14, Gerasimov2017}. The CHS are another heritage from the magnetic resonance techniques \cite{Garwood2001155}. As representative of the much broad class of adiabatic and composite pulses, CHS produce a robust inversion because for example the final state weakly depends on the pulse shape and amplitude. Within a spin or photon echo sequence, they must be applied by pairs because each CHS adds an inhomogeneous phase due to the frequency sweep. This latter can be interpreted as a sequential flipping of the inhomogeneous ensemble. Two identical CHS produce a perfect rephasing because the inhomogeneous phases induced by the CHS cancel each other \cite{minar_chirped, PascualWinter}.
CHS additionally offer{s} an advantage that is somehow underestimated. As we have just said, CHS must be appl{ied} by pairs. It means that the first echo in the ROSE sequence is also silenced because it would follow the first CHS, as opposed to the second echo which follows a pair of CHS. How much the first echo is silenced depends on the parameters of the CHS, namely the Rabi frequency and the frequency sweep. This degree of freedom should not be neglected when the phase matching conditions cannot be modified as in the cavity case in the optical or RF domain \cite{Grezes}.
To conclude about the ROSE and because of its relationship with the 2PE, it is important to question the strong pulse propagation that we pointed out as an important efficiency limitation of the 2PE (with $\pi$-pulses) by analyzing fig.\ref{fig:2PE_simul} (see appendix \ref{strong_pulse} for a more detailled discussion). In that sense as well, the CHS are superior to $\pi$-pulses. CHS are indeed very robust to propagation in absorbing media so their preserve their amplitude and frequency sweep \cite{Warren, PhysRevLett.82.3984}. CHS are not constrained by the McCall and Hahn Area Theorem (eq.\ref{area}). The latter isn't valid for frequency swept pulses \cite{Eberly:98}. This robustness to propagation can be explained qualitatively by considering the energy conservation \cite{rose}.
The different advantages of the CHS as compared to $\pi$-pulses have been studied accurately using numerical simulations in \cite{Demeter}, confirming both their versatility and robustness.
\subsection{Summary and perspectives}
We have described the variations from the well-known photon echo technique adapted for quantum storage. We haven't discussed in details the gradient echo memory scheme (GEM) \cite{hetet2008electro} (sometimes called longitudinal CRIB) which can be seen as an evolution of the CRIB protocol. The GEM is remarkable for its efficiency \cite{hetet2008electro, hedges2010efficient, hosseini2011high} allowing demonstrations in the quantum regime of operation \cite{hosseini2011unconditional}. The scheme has been enriched {by} processing functions as {a} pulse sequencer \cite{hosseini2009coherent, hosseini_jphysb}. More importantly, the GEM has been considered for RF storage in an ensemble of spins thus covering different physical realities and frequency ranges \cite{wu2010storage, zhang2015magnon}. As previously mentioned, the GEM is not covered by our formalism because the scheme couples the detuning and the position $z$. An analytical treatment is possible but is beyond the scope of our paper \cite{LongdellAnalytic}.
The specialist reader may be surprised because we did not discussed the atomic frequency comb (AFC) protocol \cite{afc} despite an undeniable series of success. The early demonstration of {weak classical field and single photon} storage \cite{usmani2010mapping, saglamyurek2011broadband, clausen2011quantum, PhysRevLett.108.190505, gundogan, PhysRevLett.115.070502, Tiranov:15, maring2017photonic} has been pushed to a remarkable level of integration \cite{saglamyurek2015quantum, PhysRevLett.115.140501, Zhong1392}. The main advantage of the AFC is a high multimode capacity \cite{afc, bonarota2011highly} which has been identified as an critical feature of the deployment of quantum repeaters \cite{collins,simon2007}. Despite a clear filiation of the AFC with the photon echo technique \cite{Mitsunaga:91}, there are also fundamental differences. For the AFC, there is no direct field to coherence mapping as discussed in section \ref{mapping}. The AFC is actually based on a population grating. Without going to much into a semantic discussion, the AFC is a descendant of the three-pulse photon echo and not the two-pulse photon echo \cite{mukamel} that we analyze in this section \ref{sec:2PE}. As a consequence, the AFC can be surprisingly linked to the {\it slow-light} protocols \cite{afc_slow} that we will discuss in the next section \ref{sec:SL}
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\section{Slow-light memories}\label{sec:SL}
Since the seminal work of Brillouin \cite{brillouin} and Sommerfeld \cite{sommerfeld}, {\it slow-light} is a fascinating subject whose impact has been significantly amplified by the popular science-fiction culture \cite{shaw}. The external control of the group velocity reappeared in the context of quantum information as a mean to store and retrieve optical {light while preserving its quantum features} \cite{EIT_Harris, fleischhauer2000dark, FLEISCHHAUER2000395}. The rest is a continuous success story that can only be embraced by review papers \cite{review_Ma_2017}.
We will start this section by deriving the Schr\"odinger-Maxwell equations used to describe the signal storage and retrieval. Our analysis is based on the following classification. We first consider the fast storage and retrieval scheme as introduced by Gorshkov {\it et al.} \cite{GorshkovII}. In other words, the storage is triggered by brief Raman $\pi$-pulses \cite{GorshkovII, legouet_raman}. We then consider the more established electromagnetically induced transparency (EIT) and the Raman schemes. In theses cases, the storage and retrieval are activated by a control field that is on or off. The difference between EIT and Raman is the control field detuning: on-resonance for the EIT scheme and off-resonance for the Raman. Both lead to very different responses of the atomic medium. In the EIT scheme, the presence of the control field produces the so-called dark atomic state. As a consequence, absorption is avoided and the medium is transparent. On the contrary, in the Raman scheme, the control beam generates an off-resonance absorption peak (Raman absorption): the medium is absorbing.
To give a common vision of the fast storage (Raman $\pi$-pulses) and the EIT/Raman schemes, we first introduce a Loren{tz}ian susceptibility response as an archetype for absorption and its counterpart the inverted-Lorentzian that describes a generic transparency window. We will define the different terms in \ref{archetypes}.
\subsection{Three-level atoms Schr\"odinger-Maxwell model}
Following the same approach as in section \ref{sec:2PE}, the pulse propagation and storage can be modeled by the Schr\"odinger-Maxwell equations in one dimension (along $z$). {We now give these equations for three level atoms.}
\subsubsection{Schr\"odinger equation for three-level atoms}
For three-level atoms, labeled $|g\rangle$, $|e\rangle$ and $|s\rangle$ for the ground, excited and spin states (see fig.\ref{fig:2level_3level}, right), the rotating-wave probability amplitudes $C_g$, $C_e$ and $C_s$ respectively are governed by the time-dependent Schr\"odinger equation similar to eq.\eqref{bloch2} \cite[eq. (13.29)]{shore2011manipulating}:
\begin{equation}gin{align}
i \partial_t \left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
C_s\\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
0 &\displaystyle \frac{\mathcal{E}^*}{2} & 0 \\
\displaystyle\frac{\mathcal{E}}{2} & -\Delta &\displaystyle \frac{\Omega}{2} \\
0 &\displaystyle \frac{\Omega^*}{2} & - \delta \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
C_s\\
\end{array}\right]
\label{bloch3}\end{align}
where $\mathcal{E}(z,t)$ and $\Omega(t)$ are the complex envelopes of the input signal and the Raman field respectively (units of Rabi frequency). If we consider the spin level $|s\rangle$ as empty, the Raman field is not attenuated (nor amplified) by the propagation so $\Omega(t)$ doesn't depend on $z$. The parameters $\Delta$ and $\delta$ are the one-photon and two-photon detunings respectively (see fig.\ref{fig:2level_3level}, right).
The atomic variables $C_g$, $C_e$ and $C_s$ depend on $z$ and $t$ for given detunings $\Delta$ and $\delta$. As in section \ref{sec:2PE}, the detunings are chosen position and time independent. Again, decay terms can be added {\it by-hand} by introducing complex detunings for $\Delta$ and $\delta$.
\subsubsection{Maxwell propagation equation}
Eqs \eqref{MB_M_hom} (homogeneous ensemble) and \eqref{MB_M_inhom} (inhomogeneous) still describe the propagation of the signal in the slowly varying envelope approximation.
The two sets of equations (\ref{bloch3}\&\ref{MB_M_hom}) or (\ref{bloch3}\&\ref{MB_M_inhom}) depending if the ensemble is homogeneous or inhomogeneous are sufficient to describe the different situations that we will consider. As already mentioned in section \ref{sec:2PE}, the equations of motion can be further simplified for weak $\mathcal{E}(z,t)$ signals (perturbative regime).
\subsubsection{Perturbative regime}
The linearisation of the Schr\"odinger-Maxwell equations (\ref{bloch3}, \ref{MB_M_hom} \&\ref{MB_M_inhom}) corresponds to the so-called perturbative regime. To the first order in perturbation, the atoms stays in the ground, $C_g \simeq 1$ because the signal is weak. The atomic evolution (eq.\ref{bloch3}) is now only given by $C_e$ and $C_s$ that we write with $\mathcal{P}\simeq C_e$ and $\mathcal{S}\simeq C_s$ to describe the optical (polarization $\mathcal{P}$) and spin ($\mathcal{S}$) excitations \cite{GorshkovII}. The atoms dynamics from eq.\eqref{bloch3} becomes:
\begin{equation}gin{align}
\partial_t \mathcal{P} &= (i\Delta-\Gamma) \mathcal{P} - i \frac{\Omega}{2} \mathcal{S} - i \frac{\mathcal{E}}{2}\label{bloch_P}\\
\partial_t \mathcal{S} &= - i \frac{\Omega^*}{2} \mathcal{P} + i \delta \mathcal{S} \label{bloch_S}
\end{align}
We have introduced the optical homogeneous linewidth $\Gamma$ that will be used later. The decay of the spin is neglected which would correspond to an infinite storage time when the excitation in shelved into the spin coherence. This is an ideal case.
The Raman field $\Omega(t)$ is unaffected by the propagation if the spin state is empty. The Raman pulse keeps its initial temporal shape so there is no differential propagation equation governing $\Omega(t)$. This a major simplification especially when a numerical integration (along $z$) is necessary.
We will only consider real envelope $\Omega(t)$ for the Raman field. Nevertheless, a complex envelope can still be used if the Raman field is chirped for example \cite{minar_chirped}.
The exact same set of equations can alternatively be derived from the density matrix formalism in the perturbative regime, the terms $\mathcal{P}$ and $\mathcal{S}$ representing the off-diagonal coherences of the $|g\rangle$-$|e\rangle$ and $|g\rangle$-$|s\rangle$ transitions respectively owing to the $C_g\simeq1$ hypothesis.
Using the polarization $ \mathcal{P}(t,\Delta)$, the Maxwell equations \eqref{MB_M_hom} and \eqref{MB_M_inhom} are rewritten as:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle{i \alpha} \Gamma \mathcal{P}(t) \label{MB_M_hom_pert}
\end{equation}
or for inhomogeneous ensembles as:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}(t,\Delta) d\Delta \label{MB_M_inhom_pert}
\end{equation}
This formalism is sufficient to describe the different situations we will consider now. The simplified perturbative set of coupled equations (\ref{bloch_P}\&\ref{bloch_S}) cannot be solved analytically when $\Omega(t)$ is time-varying, {thus} acting as a parametric driving. A numerical integration is usually necessary to fully recover the outgoing signal shape after the propagation given by eqs.\eqref{MB_M_hom_pert} or \eqref{MB_M_inhom_pert}. Simpler situations can still be examined to discuss the dispersive properties of a {\it slow-light} medium. When $\Omega(t)=\Omega$ is static, the susceptibility describing the linear propagation of the signal field $\mathcal{E}(z,t)$ can be explicitly derived. This is a very useful guide for the physical intuition.
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\subsection{Inverted-Lorentzian and Lorentzian responses: two archetypes of {slow-light}}\label{archetypes}
Before going into details, we would like to describe qualitatively two archetypal situations without specific assumption on the underlying level structure or temporal shapes of the field. From our point of view, {\it slow-light} propagation should be considered as the precursor of storage. We use the term precursor as an allusion to the work of Brillouin \cite{brillouin} and Sommerfeld \cite{sommerfeld}.
The first situation corresponds to the well-known {\it slow-light} propagation in a transparency window. More specifically, we will assume {that} the susceptibility {is} given by an inverted-Lorentzian shape. The Lorentzian should be inverted to obtain transparency and not absorption at the center. The susceptibility is defined as the proportionality constant between the frequency dependent polarization and electric field (including the vacuum permittivity $\epsilon_0$). This latter can be directly identified from the field propagation equation as we will see later in \ref{section:TW} and \ref{section:AW}.
The second situation is the complementary. A Lorentzian (non-inverted) can also be considered to produce a retarded response. This is useful guide to described certain storage protocols and revisit the concept of {\it slow-light}. The Lorentzian response naturally comes out of the Lorentz-Lorenz model when the electron is elastically bound to the nucleus when light-matter interaction is introduced to the undergraduate students. These two archetypes represent a solid basis to interpret the different protocols we will detail in section \ref{raman_stopped} and \ref{EIT_stopped}.
\subsubsection{Transparency window of an inverted-Lorentzian}\label{section:TW}
We assume that the susceptibility {is} given by an inverted-Lorentzian. This is {the} simplest case because a group delay can be explicitly derived. Whatever is the exact physical situation, the source term on the right-hand sides of eqs \eqref{MB_M_hom_pert} or \eqref{MB_M_inhom_pert} can be replaced by a linear response in the spectral domain (linear susceptibility) when $\Omega(t)$ is static. The propagation equation would read in the spectral domain \cite[p.12]{allen2012optical}
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right] \tilde{\mathcal{E}}(z,\omega) \label{propag_ILorentz} \end{equation}
where $\tilde{\mathcal{E}}(z,\omega)$ is the Fourier transform of $\mathcal{E}(z,t)$. The left-hand side simply describes the free-space propagation of the slowly varying envelope. The right-hand side is proportional to the inverted-Lorentzian susceptibility defining the complex propagation constant as:
\begin{equation}gin{equation}
\tilde{\alpha}\left(\omega\right)=-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]
\end{equation}
The different terms can be analyzed as follows. $ \frac{\alpha}{2} $ is the far-off resonance (or background) absorption coefficient for the amplitude $\tilde{\mathcal{E}}$ such as the intensity $|\tilde{\mathcal{E}}|^2$ decays exponentially with a coefficient $\alpha$ following the Bouguer-Beer-Lambert absorption law. The term $\displaystyle \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]$ represents the Lorentzian shape of a transparency window (width $\Gamma_0$) that we choose as an archetype. With this definition, the susceptibility $\chi$ can be written as $\displaystyle \chi\left(\omega\right)=-\frac{2i}{k} \tilde{\alpha}\left(\omega\right)$ where $k$ is the wavevector \footnote{With our definitions, the real part of the propagation constant $\tilde{\alpha}$ gives the absorption and the imaginary part, the dispersion. For the susceptibility, this is the other way around.}.
At the center $\omega=0$, there is no absorption (complete transparency). We choose a complex Lorentzian $\displaystyle \frac{1}{1+i\omega/\Gamma_0}$ and not a real one $\displaystyle \frac{1}{1+\omega^2/\Gamma_0^2}$ because the complex Lorentzian satisfies {\it de facto} the Kramers-Kronig relation so we implicitly respect the causality. The propagation within the transparency window is given by a first-order expansion of the susceptibility when $\omega \ll \Gamma_0$ leading to
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega) \simeq \displaystyle-\frac{\alpha}{2} i\frac{\omega}{\Gamma_0} \tilde{\mathcal{E}}(z,\omega) \end{equation}
and after integration over the propagation distance $L$
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) \simeq \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{i \omega \alpha L }{2\Gamma_0} \right) \end{equation}
or equivalently in the time domain
\begin{equation}gin{equation} \label{eq:nunn2} \mathcal{E}(L,t) \simeq \mathcal{E}(0,t-\frac{L }{c}-\frac{\alpha L}{2 \Gamma_0}) \end{equation}
where $\displaystyle \frac{L }{c}+\frac{d}{2 \Gamma_0}$ is the group delay with the optical depth $d=\alpha L$. If the incoming pulse bandwidth fits the transparency window or in other words if the pulse is sufficiently long, the pulse is simply delayed by $\displaystyle \frac{d}{2 \Gamma_0}$. This latter defines the group delay.
Shorter pulses are distorted and partially absorbed when the bandwidth extends beyond the transparency window. In that case, eq.\eqref{propag_ILorentz} can be integrated analytically to give the general formal solution:
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) = \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{d}{2} \frac{i \omega}{\Gamma_0+i\omega}\right) \label{SL_TW} \end{equation}
The outgoing pulse shape $\mathcal{E}(L,t) $ is given by the inverse Fourier transform of $ \tilde{ \mathcal{E}}(L,\omega)$. As an example, we plot the outgoing pulse in fig.\ref{fig:SL_ILorentz} for a Gaussian input $\displaystyle \mathcal{E}(0,t)= \exp \left(-\frac{t^2}{2\sigma^2}\right)$. We choose $\Gamma_0=1$ and a pulse duration $\sigma =\displaystyle \frac{d}{2 \Gamma_0}$ corresponding to the expected group delay. We take $d=20$ for the optical depth, which corresponds to realistic experimental situations.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.65\linewidth]{SL_ILorentz.eps}}
\caption{{\it Slow-light} in a Lorentzian transparency window. Top: real and imaginary part of the propagation constant $\tilde{\alpha}\left(\omega\right)$. The real part represents the absorption and the imaginary part the refractive index (dispersion) whose slope is the group delay. The shaded area corresponds to the {\it slow-light} region, the positive slope of the imaginary part leads to a positive group delay. Bottom: {\it Slow-light} propagation of a Gaussian incoming pulse (in blue) producing a delayed output pulse (in red) calculated from eq.\eqref{SL_TW}.}
\label{fig:SL_ILorentz}
\end{figure}
The outgoing pulse is essentially delayed by $\displaystyle \frac{d}{2 \Gamma_0}=10$ and only weakly absorbed through the propagation. A longer pulse would lead to less absorption but the input and output would be much less separated. As we will see later, this point is critical for {\it slow-light} storage protocols.
\subsubsection{Dispersion of a Lorentzian}\label{section:AW}
We now consider a Lorentzian as a complementary situation. This may sound surprising for the reader familiar with the EIT transparency window. {However,} the Lorentzian is a useful reference to interpret the Raman memory {that will be discussed in section \ref{Raman}.
We {consider a} propagation constant given by
\begin{equation}gin{equation}
\tilde{\alpha}\left(\omega\right)=-\frac{\alpha}{2} \frac{1}{1+i\omega/\Gamma_0}
\end{equation}
This is a quite simple case corresponding to the transmission of an homogeneous ensemble of dipoles. To take the terminology of the previous case, one could speak of an absorption window as opposed to a transparency window. To follow up the analogy, there is no {\it slow-light} at the center of an absorption profile. The susceptibility is inverted thus leading to {\it fast-light} (negative group delay). A retarded response can still be expected but on the wings (off-resonance) of the absorption profile. As represented on fig.\ref{fig:SL_Lorentz}, the slope is negative at the center ({\it fast-light}) but it changes sign out of resonance leading to a distorted version of {\it slow-light}. Distortion are indeed expected because the dispersion cannot be considered as linear. Still, what comes out of the medium after the incoming pulse can be interpreted as a precursor for light storage.
By inverted analogy with the previous case, the propagation can be solved to the first order when the pulse bandwidth is much larger than the absorption profile (off-resonant excitation of the wings). The Lorentzian $\displaystyle \frac{1}{1+i\omega/\Gamma_0}$ simplifies to the first order in $\displaystyle \frac{\Gamma_0}{i\omega}$ leading to the solution in the spectral domain:
\begin{equation}gin{equation} \label{eq:propag3} \tilde{ \mathcal{E}}(L,\omega) \simeq \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{\alpha L \Gamma_0}{2i \omega} \right) \end{equation}
or alternatively in the time domain
\begin{equation}gin{equation} \mathcal{E}(L,t) \simeq \mathcal{E}(0,t) \ast F(L,t) \label{FID_convol} \end{equation}
where $F(L,t)$ is the impulse response convoluting ($\ast$) the incoming pulse shape and analytically given by \cite{bateman1954tables}:
\begin{equation}gin{equation}\label{eq:FID}
F\left(L,t\right) = \delta_{t=0} - {\alpha L \Gamma_0} \frac{J_1\left(\sqrt{2d \Gamma_0 t}\right)}{\sqrt{2d \Gamma_0 t}} \mbox{ for t$>$0 and 0 elsewhere} \end{equation}
$J_1$ is the Bessel function of the first kind of order 1 with the optical depth $d=\alpha L$. $\delta_{t=0}$ is the Dirac peak. The time $\displaystyle \frac{1}{d \Gamma_0}$ appears as a typical delay due to propagation. The output shape will be distorted by the strong oscillations of the Bessel function. This can be investigated by considering the following numerical example without first order expansion. The output shape is indeed more generally given by the inverse Fourier transform of the integrated form:
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) = \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{d }{2} \frac{\Gamma_0}{\Gamma_0+i\omega}\right) \label{SL_TA} \end{equation}
Again we plot the outgoing pulse in fig.\ref{fig:SL_Lorentz} for a Gaussian input $\displaystyle \mathcal{E}(0,t)= \exp \left(-\frac{t^2}{2\sigma^2}\right)$ whose duration is now $\sigma =\displaystyle\frac{1}{d \Gamma_0}$ ($\Gamma_0=1$) corresponding to the expected generalized group delay. As before, the optical depth is $d=20$.
Two lobes appear at the output (fig.\ref{fig:SL_Lorentz}) as expected from the approximated expression eq.\eqref{FID_convol} involving the oscillating Bessel function. Still, a significant part of the incoming pulse is retarded in the general sense whatever is the exact outgoing shape.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.65\linewidth]{SL_Lorentz.eps}}
\caption{{\it Slow-light} from a Lorentzian absorption window. Top: real and imaginary part of the propagation constant $\tilde{\alpha}\left(\omega\right)$. The shaded area corresponds to the {\it slow-light} region (positive group delay). Bottom: {\it Slow-light} propagation of a gaussian incoming pulse (in blue) producing a retarded output pulse (in red) calculated from eq.\eqref{SL_TA}.}
\label{fig:SL_Lorentz}
\end{figure}
As will see now, what is retarded can be stored.
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\subsubsection{Dispersion of a Lorentzian}\label{section:AW}
We now consider a Lorentzian as a complementary situation. This may sound surprising for the reader familiar with the EIT transparency window. {However,} the Lorentzian is a useful reference to interpret the Raman memory {that will be discussed in section \ref{Raman}.
We {consider a} propagation constant given by
\begin{equation}gin{equation}
\tilde{\alpha}\left(\omega\right)=-\frac{\alpha}{2} \frac{1}{1+i\omega/\Gamma_0}
\end{equation}
This is a quite simple case corresponding to the transmission of an homogeneous ensemble of dipoles. To take the terminology of the previous case, one could speak of an absorption window as opposed to a transparency window. To follow up the analogy, there is no {\it slow-light} at the center of an absorption profile. The susceptibility is inverted thus leading to {\it fast-light} (negative group delay). A retarded response can still be expected but on the wings (off-resonance) of the absorption profile. As represented on fig.\ref{fig:SL_Lorentz}, the slope is negative at the center ({\it fast-light}) but it changes sign out of resonance leading to a distorted version of {\it slow-light}. Distortion are indeed expected because the dispersion cannot be considered as linear. Still, what comes out of the medium after the incoming pulse can be interpreted as a precursor for light storage.
By inverted analogy with the previous case, the propagation can be solved to the first order when the pulse bandwidth is much larger than the absorption profile (off-resonant excitation of the wings). The Lorentzian $\displaystyle \frac{1}{1+i\omega/\Gamma_0}$ simplifies to the first order in $\displaystyle \frac{\Gamma_0}{i\omega}$ leading to the solution in the spectral domain:
\begin{equation}gin{equation} \label{eq:propag3} \tilde{ \mathcal{E}}(L,\omega) \simeq \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{\alpha L \Gamma_0}{2i \omega} \right) \end{equation}
or alternatively in the time domain
\begin{equation}gin{equation} \mathcal{E}(L,t) \simeq \mathcal{E}(0,t) \ast F(L,t) \label{FID_convol} \end{equation}
where $F(L,t)$ is the impulse response convoluting ($\ast$) the incoming pulse shape and analytically given by \cite{bateman1954tables}:
\begin{equation}gin{equation}\label{eq:FID}
F\left(L,t\right) = \delta_{t=0} - {\alpha L \Gamma_0} \frac{J_1\left(\sqrt{2d \Gamma_0 t}\right)}{\sqrt{2d \Gamma_0 t}} \mbox{ for t$>$0 and 0 elsewhere} \end{equation}
$J_1$ is the Bessel function of the first kind of order 1 with the optical depth $d=\alpha L$. $\delta_{t=0}$ is the Dirac peak. The time $\displaystyle \frac{1}{d \Gamma_0}$ appears as a typical delay due to propagation. The output shape will be distorted by the strong oscillations of the Bessel function. This can be investigated by considering the following numerical example without first order expansion. The output shape is indeed more generally given by the inverse Fourier transform of the integrated form:
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) = \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{d }{2} \frac{\Gamma_0}{\Gamma_0+i\omega}\right) \label{SL_TA} \end{equation}
Again we plot the outgoing pulse in fig.\ref{fig:SL_Lorentz} for a Gaussian input $\displaystyle \mathcal{E}(0,t)= \exp \left(-\frac{t^2}{2\sigma^2}\right)$ whose duration is now $\sigma =\displaystyle\frac{1}{d \Gamma_0}$ ($\Gamma_0=1$) corresponding to the expected generalized group delay. As before, the optical depth is $d=20$.
Two lobes appear at the output (fig.\ref{fig:SL_Lorentz}) as expected from the approximated expression eq.\eqref{FID_convol} involving the oscillating Bessel function. Still, a significant part of the incoming pulse is retarded in the general sense whatever is the exact outgoing shape.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.65\linewidth]{SL_Lorentz.eps}}
\caption{{\it Slow-light} from a Lorentzian absorption window. Top: real and imaginary part of the propagation constant $\tilde{\alpha}\left(\omega\right)$. The shaded area corresponds to the {\it slow-light} region (positive group delay). Bottom: {\it Slow-light} propagation of a gaussian incoming pulse (in blue) producing a retarded output pulse (in red) calculated from eq.\eqref{SL_TA}.}
\label{fig:SL_Lorentz}
\end{figure}
As will see now, what is retarded can be stored.
\subsubsection{A retarded response as a precursor for storage}
{\it Slow-light} is a precursor of storage called {\it stopped-light} in that case. The transition from {\it slow} to {\it stopped-light} is summarized in fig.\ref{fig:SL_Lorentz_ILorentz_shaded}.
When input and output are well separated in time, storage is possible in principle. If we look at the standard situation of {\it slow-light} in a transparency window (fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, top), we choose a frontier between input and output at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$. At this given moment, most of the output pulse has entered the atomic medium. There is only a small fraction of the input pulse (blue shaded area) that leaks out. This part will be lost. Concerning the output pulse, the red shaded area (subtracted from the blue area) is essentially contained inside the medium and {\it de facto} stored into the atomic excitation \cite{Shakhmuratov, ChaneliereHBSM}. The same qualitative description also applies to the retarded response from a Lorentzian absorption window (fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, bottom). Storage can be expected as well but at the price of temporal shape distortion.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.7\linewidth]{SL_Lorentz_ILorentz_shaded.eps}}
\caption{Top: {\it Slow-light} in a transparency window as in fig. \ref{fig:SL_ILorentz}. The shaded area after half of the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$ represents the separation between the input and the outgoing pulses. Bottom: Retarded response from a Lorentzian absorption window as in fig. \ref{fig:SL_Lorentz}. We choose for the separation between input and output the expected generalized group delay $\displaystyle\frac{1}{d \Gamma_0}=.05$.}
\label{fig:SL_Lorentz_ILorentz_shaded}
\end{figure}
Following our interpretation, as soon as input and output are well separated, there is a moment when a f{r}action of the light is contained in the atomic excitation. This fraction defines the storage efficiency. The transition from {\it slow} to {\it stopped-light} requires to detail the specific storage protocols by giving a physical reality to the (inverted-)Lorenzian susceptibility. {\it Slow-light} ensures that the optical excitation is transiently contained in the atomic medium. For {permanent} storage {and on-demand readout}, it is necessary to act dynamically on the atomic excitation as we will see now. More precisely, the shelving of the excitation into the spin (by a brief Raman $\pi$-pulses or by switching off the control field as we will see in \ref{raman_stopped} and \ref{EIT_stopped} respectively) prevents the radiation of the retarded response. The excitation is trapped in the atomic ensemble. The evolution is resumed at the retrieval stage by the reversed operation (by a second brief Raman pulses or by switching on the control field).
Before going into details of the storage schemes, we briefly show that the correct orders of magnitude for the efficiencies can be derived from our simplistic vision. From fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, we can roughly evaluate the efficiency by subtracting the blue from the red area assuming the incoming energy (integral of the incoming pulse) is one. We find for {\it slow-light} in a transparency window (inverted-Lorentzian profile) a potential efficiency of 43\% and for the retarded response of an absorption window (Lorentzian profile) 32\%.
We will keep these numbers as points of comparison for specific protocols that we will first explicitly connect to the {\it slow-light} propagation from an inverted-Lorentzian or a Lorentzian and then numerically simulate with the previously established Schr\"odinger-Maxwell equations.
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\subsection{Fast storage and retrieval with brief Raman $\pi$-pulses}\label{raman_stopped}
Our approach is based on the fact that {\it slow-light} is associated with the transient storage of the incoming pulse into the atomic excitation. A simple method to store {more} permanently the excitation is to convert instantaneously the optical excitation into a spin wave. This can be done by a Raman $\pi$-pulse as proposed in different protocols. We will now go into details and properly define the level structure and the temporal sequence required to implement the previously discussed archetypes (sections \ref{section:TW} and \ref{section:AW}). We will consider two specific protocols: the spectral hole memory and the free induction decay memory proposed in \cite{Lauro1} and \cite{Vivoli} respectively.
\subsubsection{Spectral hole memory}\label{SHOME}
The spectral hole memory has been proposed by Lauro {\it et al.} in \cite{Lauro1} and partially investigated experimentally in \cite{Lauro2}. The protocol has been successfully implemented in \cite{SHOME} at the single photon level with a quite promising efficiency of 31\%. An inhomogeneously broaden ensemble is first considered. A spectral hole is then burnt into the inhomogeneous distribution. This situation is realistic and corresponds to rare-earth doped crystals for which the spectral hole burning mechanism, as spectroscopic tool, can be efficiently used to sculpt the absorption profile \cite{liu2006spectroscopic}. When the hole profile is Lorenztian, the propagation of a weak signal pulse precisely corresponds to the situation \ref{section:TW} as we will see now.
The atomic evolution is described by eqs.(\ref{bloch_P}\&\ref{bloch_S}) and the propagation by eq.\eqref{MB_M_inhom_pert}. The signal $\mathcal{E}(z,t)$ propagates initially through the atomic distribution described by
\begin{equation}gin{equation}
g\left(\Delta\right)=\left[ 1- \frac{1}{1+\left(\Delta/\Gamma_0\right)^2} \right]\label{g_shome}
\end{equation}
where $\Gamma_0$ is the spectral hole width.
The Raman field is initially off and is only applied for the rapid conversion into the spin wave. When the Raman field is off, the evolution eq.\eqref{bloch_P} reads as $\displaystyle \partial_t \mathcal{P} = (i\Delta-\Gamma) \mathcal{P} - i \frac{\mathcal{E}}{2}$. The coherence lifetime $1/\Gamma$ (inverse of the homogeneous linewidth) is assumed to be much longer than the time of the experiment such as in the spectral domain we write in the limit $\Gamma \rightarrow 0$
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ \tilde{\mathcal{E}}(z,\omega)}{2\left(\Delta-\omega\right)} \end{equation}
So the propagation reads as
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta \end{equation}
The term $\displaystyle \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta$ represents the susceptibility. The integral over $\Delta$ ensures that the Kramers-Kroning relations are satisfied. This last term is then given by the Hilbert transform of the distribution $g\left(\Delta\right)$ so we have $\displaystyle \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta =\left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]$. The propagation of the signal is indeed given by eq.\eqref{propag_ILorentz} as described in section \ref{section:TW} and as represented in figs.\ref{fig:SL_ILorentz} and \ref{fig:SL_Lorentz_ILorentz_shaded} (top). The delayed pulse (or at least the fraction which is sufficiently separated from the input) can be stored as represented by shaded areas in fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (top). As proposed in \cite{Lauro1}, a Raman $\pi$-pulse can be used to shelve the optical excitation into the spin. A second Raman $\pi$-pulse triggers the retrieval. They are applied on resonance ($|s\rangle$-$|e\rangle$ transition) so $\delta=0$ in eq.\eqref{bloch_S}.
When the input and the output overlap as in many realistic situations or in other words when the signal cannot be fully compressed spatially into the medium, the storage step cannot be solved analytically. A numerical simulation of the Schr\"odinger-Maxwell equations is necessary (eqs.\ref{bloch_P}\&\ref{bloch_S} with $\Gamma=0$ and $\delta=0$ and eq.\eqref{MB_M_inhom_pert} for the propagation). For a given inhomogeneous detuning $\Delta$, we calculate the atomic evolution eqs.(\ref{bloch_P}\&\ref{bloch_S}) by using a fourth-order Runge-Kutta method. After summing over the inhomogeneous broadening, the output pulse is given by integrating eq.\eqref{MB_M_inhom_pert} along $z$ using the Euler method. A good test for the numerical simulation is to calculate the output pulse without Raman pulses and compare it to the analytic expression from the Fourier transform of eq.\eqref{SL_TW}.
The Raman $\pi$-pulses defined by $\Omega(t)$ are taken as two Gaussian pulses whose area is $\pi$. Following the insight of fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (top), we choose to apply the first Raman pulse at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$. The second Raman pulse is applied later to trigger the retrieval. The result of the storage and retrieval sequence is presented in fig.\ref{fig:plot_outputIO_SHOME}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_SHOME.eps}}
\caption{Spectral hole memory protocol. Top: Absorption profile from the inhomogeneous distribution $g$ defined by eq.\eqref{g_shome}. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference when there is no Raman pulse. Bottom: Two Raman $\pi$-pulses. The first one is applied at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$ and the second later on to trigger the retrieval.
}
\label{fig:plot_outputIO_SHOME}
\end{figure}
As parameters for the simulation, we choose the same as in \ref{section:TW} meaning $\Gamma_0=1$, an optical depth of $d=20$ and $\sigma =\displaystyle \frac{d}{2 \Gamma_0}=10$ for the incoming pulse duration. The Raman pulses should be ideally short to uniformly cover the signal excitation bandwidth. In our case, we choose Gaussian Raman pulses with a duration $\sigma_\pi=1$ (ten times shorter than the signal).
In fig.\ref{fig:plot_outputIO_SHOME} (middle), we clearly see that the first Raman pulse somehow clips the {\it slow-light} pulse corresponding to the shelving of the optical excitation into the spin wave. At this moment, since part of the input pulse is still present, a small replica is generated leaving the medium at time $20$ in our units. The second Raman $\pi$-pulse (at time $60$) triggers the retrieval that we shaded in pale red. A realistic storage situation cannot be fully described by our qualitative picture in fig.\ref{fig:SL_Lorentz_ILorentz_shaded} where the {\it slow-light} signal would be clipped, frozen, delayed and retrieved later on. The complex propagation of clipped Gaussian excitations in the medium can only be accurately embraced by a numerical simulation. The naive picture gives nonetheless a qualitative guideline to understand the storage. A quantitative analysis can be performed by evaluating the stored energy corresponding to the pale red shaded area under the intensity curve. From the simulation, we obtain 36\% to be compared with the 43\% obtained from fig.\ref{fig:SL_Lorentz_ILorentz_shaded}. The agreement is satisfying given the numerical uncertainties and the complexity of the propagation process when the Raman pulses are applied. We now turn to the complementary situation described in section \ref{section:AW} by following the same procedure.
\subsubsection{Free induction decay memory} \label{FID}
Free induction decay memory to take the terminology of the article by Caprara Vivoli {\it et al.} \cite{Vivoli} has not been yet implemented in practice despite a connection with the extensively studied {\it slow-light} protocols. The situation actually corresponds to our description in section \ref{section:AW} where the response of a Lorenztian to a pulsed excitation is considered. This response has been analyzed as a generalization of the free induction decay phenomenon (FID) by Caprara Vivoli {\it et al.} \cite{Vivoli}. The FID is usually observed in low absorption sample after a brief excitation. The analysis in terms of FID is perfectly valid. The response that we considered with eq.\eqref{eq:FID} with a first order expansion of the susceptibility falls into this framework. We analyze the same situation in different terms recovering the same reality. The excitation produces a retarded response that we consider as a generalized version of {\it slow-light}. This semantically connects {\it slow-light} and FID in the context of optical storage.
For the FID memory, the transition can be inhomogeneously or homogeneously broaden{ed}. Both lead to the same susceptibility. We assume the medium homogenous with a linewidth $\Gamma$ thus simplifying the analysis and the numerical simulation, the propagation being given by eq.\eqref{MB_M_hom_pert}.
As in the spectral hole memory, the Raman field is initially off and serves as a rapid conversion into the spin wave by the application of a $\pi$-pulse. When the Raman field is off, the evolution (eq.\ref{bloch_P}) reads as $\displaystyle \partial_t \mathcal{P} = -\Gamma \mathcal{P} - i \frac{\mathcal{E}}{2}$. The signal is directly applied on resonance so $\Delta=0$. We then obtain for the polarization \begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega+\Gamma\right)} \end{equation}
and the propagation
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{\Gamma}{i\omega+\Gamma} \end{equation} whose solution is indeed given by eq.\eqref{SL_TA}. The output pulse is distorted and globally affected by a typical delay $\displaystyle \frac{1}{d \Gamma}$. A first Raman $\pi$-pulse can be applied at this moment. A second Raman $\pi$-pulse triggers the retrieval. As in the spectral hole memory (section \ref{SHOME}), they are applied on resonance so $\delta=0$ in eq.\eqref{bloch_S}.
The complete protocol (when Raman pulses are applied) can only be simulated numerically from the Schr\"odinger-Maxwell equations (eqs.\ref{bloch_P}\&\ref{bloch_S} with $\Delta=0$ and $\delta=0$ and eq.\eqref{MB_M_hom_pert} for the propagation in an homogeneous sample). Following fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (bottom), we choose to apply the first Raman pulse at the generalized group delay $\displaystyle \frac{1}{d \Gamma}$. The second Raman pulse is applied later on to trigger the retrieval.
For the simulation, we again choose the parameters used in section \ref{section:AW} namely a linewdith $\Gamma=1$ and a signal pulse duration of $\sigma =\displaystyle\frac{1}{d \Gamma}=0.05$ corresponding to the expected generalized group delay for an optical depth $d=20$. The Raman pulses have a duration $\sigma_\pi=0.005$ (ten times shorter than the signal) and a $\pi$-area. The result is presented in fig.\ref{fig:plot_outputIO_FID}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_FID.eps}}
\caption{Free induction decay memory. Top: Lorentzian absorption profile from an homogeneous sample. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference when there is no Raman pulse. Bottom: Two Raman $\pi$-pulses. The first one is applied at $\displaystyle \frac{1}{d \Gamma}=0.05$ and the second later on to trigger the retrieval.}
\label{fig:plot_outputIO_FID}
\end{figure}
We retrieve the tendencies of the spectral hole memory. The first Raman pulse clips the {\it slow-light} pulse by storing the excitation into the spin state. As opposed to the propagation in the spectral hole, there is no replica after the first Raman pulse. This replica is strongly attenuated (slightly visible in fig.\ref{fig:plot_outputIO_FID}) because it propagates through the absorption window. We trigger the retrieval at time $0.8$ by a second Raman $\pi$-pulse. The temporal output shape cannot be compared to a clipped version of the input or the {\it slow-light} pulse. This situation is clearly more complex than the spectral hole memory. That being said, the resemblance of the output shape with a exponential decay somehow a posteriori justifies the term FID for this memory scheme. The red pale shaded area represents an efficiency of 42\% with respect to the input pulse energy. This numerical result has to be compared with 32\% obtained from fig.\ref{fig:SL_Lorentz_ILorentz_shaded}. The agreement is not satisfying even if it is difficult to have a clear physical vision of the pulse distortion induced by the propagation at large optical depth. The order of magnitude is nevertheless correct.
The FID protocol can be optimally implemented by using an exponential rising pulse for the incoming signal (instead of a Gaussian in fig.\ref{fig:plot_outputIO_FID}, middle) as analyzed in the reference paper \cite{Vivoli}. In that case, input (rising exponential) and output (decaying exponential) pulse shapes are time-reversed corresponding to the optimization procedures defined in \cite{GorshkovII, GorshkovPRL} and implemented in the EIT/Raman memories \cite{Novikova, nunnMultimode, zhou2012optimal}
Starting from two representative situations in \ref{section:TW} and \ref{section:AW} where the dispersion produces a retarded response from the medium, we have analyzed two related protocols in \ref{SHOME} and \ref{FID} that qualitatively corresponds to the storage of this delayed response. Except in a recent implementation \cite{SHOME}, these protocols have not been much considered in practice despite a clear connection with the archetypal propagation through the Lorentzian susceptibility of an atomic medium. On the contrary, electromagnetically induced transparency and Raman schemes are well-known and extensively studied experimentally. We will show now that they follow the exact same classification thus enriching our comparative analysis.
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\subsection{Electromagnetically induced transparency and Raman schemes}\label{EIT_stopped}
Starting from two pioneer realizations \cite{phillips2001storage, liu2001observation}, the implementation of the electromagnetically induced transparency (EIT) scheme has been continuously active in the prospect of quantum storage. As opposed to the spectral hole (section \ref{SHOME}) and the free induction decay (section \ref{FID}) memories and recalling to the reader the main difference, EIT is not based on the transient excitation of the optical transition that is rapidly transfered into the spin by a Raman $\pi$-pulse. In EIT, the direct optical excitation is avoided by precisely using the so-called dark state in a $\Lambda$-system \cite{fleischhauer2000dark, FLEISCHHAUER2000395}. Practically, a control field is initially applied on the Raman transition to obtain {\it slow-light} from the $\Lambda$-system susceptibility\footnote{Inversely, for the spectral hole in \ref{SHOME} and the free induction decay in \ref{FID} memory, the Raman field is initially off.}. As a first cousin, the Raman memory scheme has been proposed and realized afterward \cite{nunnPRA,nunnNat}. EIT and Raman memories are structurally related by a common $\Lambda$-system which is weakly excited by the signal on one branch and controlled by a strong laser on the Raman branch (see fig.\ref{fig:2level_3level}). The main difference comes from the excited state detuning. For EIT scheme, the control field is on resonance. For the Raman scheme, the control field is off resonance. As we will see now, these two situations actually corresponds to the archetypal dispersive profiles described in \ref{section:TW} and \ref{section:AW} respectively.
\subsubsection{Electromagnetically induced transparency memory}\label{EIT}
The atomic susceptibility in a $\Lambda$-system is derived from eqs.(\ref{bloch_P}\&\ref{bloch_S}). Initially, the Raman field $\Omega$ is on and assumed constant in time. In EIT, the signal and control fields are on resonance so $\Delta=\delta=0$. The medium is assumed homogeneous even if the calculation can be extended to the inhomogeneously broaden systems \cite{kuznetsova2002atomic}. The propagation is here given by eq.\eqref{MB_M_hom_pert}.
In the spectral domain, eqs.(\ref{bloch_P}\&\ref{bloch_S}) read as
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega+\Gamma-\displaystyle i\frac{\Omega^2}{4\omega}\right)} \end{equation}
We have assumed the control field {to be} real so the intensity is written as $\Omega^2$ which can be generalized to $\Omega^*\Omega$ for complex values (chirped Raman pulses for example).
The linear susceptibility for the signal field is defined by the propagation equation in the spectral domain
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{\Gamma}{i\omega+\Gamma - \displaystyle i\frac{\Omega^2}{4\omega} } \label{propag_EIT}\end{equation}
The term $i \displaystyle \frac{\Omega^2}{4\omega}$ induces the transparency when the control field is applied. Without control, the susceptibility is Lorentzian and the signal would be absorbed following the Bouguer-Beer-Lambert absorption law (eq.\ref{bouguer}). On the contrary, when the control field is on, the susceptibility is zero when $\omega \rightarrow 0$. This corresponds to the resonance condition because we assumed $\Delta=\delta=0$. The analysis can be further simplified by considering a first order expansion within the transparency window.
The width of the transparency window is $\ensuremath{\Gamma_\mathrm{EIT}}=\displaystyle \frac{\Omega^2}{4\Gamma}$ which is usually much narrower than $\Gamma$. So, in the limit $\omega \ll \ensuremath{\Gamma_\mathrm{EIT}} \ll \Gamma$, the propagation constant reads as
\begin{equation}gin{equation} \displaystyle-\frac{\alpha}{2} \frac{\Gamma}{i\omega+\Gamma - \displaystyle i\frac{\Omega^2}{4\omega} } \simeq \displaystyle-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\ensuremath{\Gamma_\mathrm{EIT}}} \right] \end{equation}
The EIT window is locally an inverted-Lorentzian that we have analyzed in \ref{section:TW}. The {\it slow-light} propagation is precisely due to the presence of the control field. The so-called dark state corresponds to a direct spin wave excitation whose radiation is mediated by the control field. The storage simply requires the extinction of the control field. The excitation is then frozen in the non-radiating Raman coherence because of the absence of control. The retrieval is triggered by switching the control back on.
The {\it stopped-light} experimental sequence can be simulated numerically from eqs.(\ref{bloch_P}\&\ref{bloch_S}) and eq.\eqref{MB_M_hom_pert}. For the parameters, we choose the same as in \ref{section:TW} and \ref{SHOME}, meaning $\ensuremath{\Gamma_\mathrm{EIT}}=1$ so the width of the inverted-Lorentzian is $1$. We opt for $\Omega=4$ and $\Gamma=4$ so the condition $\ensuremath{\Gamma_\mathrm{EIT}} \ll \Gamma$ is vaguely satisfied. Again the optical depth is $d=20$ and $\sigma =\displaystyle \frac{d}{2 \ensuremath{\Gamma_\mathrm{EIT}}}=10$ is the incoming pulse duration. At time $\displaystyle \frac{d}{4\ensuremath{\Gamma_\mathrm{EIT}}}=5$, half the group delay, the control field is switched off ($\Omega=0$). The result is plotted in fig.\ref{fig:plot_outputIO_EIT}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_EIT.eps}}
\caption{Electromagnetically induced transparency memory. Top: EIT absorption profile. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference from eq.\eqref{propag_EIT} when the control field is always on. Bottom: The control is initially on with $\Omega^2=16$. It is switched off at half of the group delay $\displaystyle \frac{d}{4 \ensuremath{\Gamma_\mathrm{EIT}}}=5$ and back on later to trigger the retrieval.}
\label{fig:plot_outputIO_EIT}
\end{figure}
Although the condition $\ensuremath{\Gamma_\mathrm{EIT}}=1 \ll \Gamma=4$ is only roughly satisfied so the absorption profile is not a pure inverted-Lorenzian, this has a minor influence on the {\it slow} and {\it stopped-light} pulses. The resemblance with fig.\ref{fig:plot_outputIO_SHOME} is striking even if the spectral hole and EIT memories cover different physical realities. From fig.\ref{fig:plot_outputIO_EIT}, we can estimate the efficiency (red pale area) to 42\% thus retrieving the same expected efficiency as the spectral hole memory. One difference between fig.\ref{fig:plot_outputIO_SHOME}(middle) and fig.\ref{fig:plot_outputIO_EIT}(middle) is worth being commented: there is not {\it slow-light} replica after the time $\displaystyle \frac{d}{4 \ensuremath{\Gamma_\mathrm{EIT}}}=5$ for the EIT situation. This replica is absorbed in that case because when the control field is switched off, the absorption is fully restored. The presence or the absence of replicas does not change the efficiency because they correspond to a fraction of the incoming pulse that is not compressed in the medium. This leaks out and is lost anyway.
We will now complete our picture by considering the Raman memory and emphasize the resemblance with the free induction decay discussed in \ref{FID}.
\subsubsection{Raman memory}\label{Raman}
The Raman memory scheme is based on the same $\Lambda$-structure when a control field is applied far off-resonance on the Raman branch \cite{nunnPRA,nunnNat, sheremet} (see fig.\ref{fig:2level_3level}). The condition $\Delta \gg \Gamma$ defines literally the Raman condition as opposed to EIT where the control is on resonance ($\Delta=0$). The absorption profile exhibits the so-called Raman absorption peak. This Lorentzian profile is the basis for a retarded response that we introduced in \ref{section:AW}. We first verify that the far off-resonance excitation of the control leads to a Lorenztian susceptibility for the signal. As in the EIT case (see \ref{EIT}), the atomic evolution in a $\Lambda$-system is given by eqs.(\ref{bloch_P}\&\ref{bloch_S}) and the propagation by eq.\eqref{MB_M_hom_pert}.
The polarization is
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega-i\Delta+\Gamma-\displaystyle i\frac{\Omega^2}{4(\omega-\delta)}\right)} \end{equation}
The two-photon detuning $\delta$ is not zero in that case because the Raman absorption peak in shifted by the AC-Stark shift (light shift). The signal pulse has to be detuned by $\delta=\displaystyle \frac{\Omega^2}{4\Delta}$, the light-shift, to be centered on the Raman absorption peak. Following the same approach as in the EIT case, the analysis can be simplified by a first order expansion is $\omega$. Assuming the incoming pulse bandwidth $\omega$ smaller than the light shift $\delta,$ the latter being smaller than the detuning $\Delta$, that is $\omega \ll \delta \ll \Delta$, the propagation constant reads to the first order in $\omega$ as
\begin{equation}gin{equation}
\displaystyle-\frac{\alpha}{2} \frac{\Gamma}{i\omega-i\Delta+\Gamma - \displaystyle i\frac{\Omega^2}{4(\omega-\delta)} } \simeq \displaystyle-\frac{\alpha}{2} \frac{\ensuremath{\Gamma_\mathrm{R}}}{\ensuremath{\Gamma_\mathrm{R}}+i\omega} \label{Raman_susceptibility}
\end{equation}
where $\displaystyle \ensuremath{\Gamma_\mathrm{R}}=\frac{\Omega^2\Gamma}{4\Delta^2}$ is the width of the Raman absorption profile. This Lorentzian absorption profile can be used for storage as discussed in \ref{section:AW} and \ref{FID}. As in the EIT case, the storage is triggered by the extinction of the Raman control field. To fully exploit the analogy with \ref{section:AW} and \ref{FID}, we will choose $\ensuremath{\Gamma_\mathrm{R}}=1$. To satisfy the far off resonance Raman condition, we choose $\Gamma=10$ and $\Delta=1000$ thus imposing $\Omega=200\sqrt{10}$ and $\delta=100$. We run a numerical simulation of eqs.(\ref{bloch_P}\&\ref{bloch_S}) and eq.\eqref{MB_M_hom_pert} with a Gaussian incoming pulse whose duration is again $\sigma =\displaystyle\frac{1}{d \Gamma_R}=0.05$ and with an optical depth $d=20$. The result is presented in fig.\ref{fig:plot_outputIO_Raman} where the control Raman field is switched off at time $\displaystyle\frac{1}{d \Gamma_R}=0.05$ (the typical delay) and switched back on later to trigger the retrieval.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_Raman.eps}}
\caption{Raman memory. Top: Raman absorption profile. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulsed (dashed red) as a reference from eq. \eqref{Raman_susceptibility} when the Raman control field is always on. Bottom: The control is initially on with $\Omega=200\sqrt{10}$ and is switched later on to trigger the retrieval.}
\label{fig:plot_outputIO_Raman}
\end{figure}
The resemblance with fig.\ref{fig:plot_outputIO_FID} is noticeable. Transient rapid oscillations appears when the control is abruptly switched, this is a manifestation of the light-shift. Without surprise, the expected efficiency (red-pale area) is 42\% as the free induction decay memory with the same intensive parameters (see \ref{FID}).
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\subsection{Summary and perspectives}
We have given in this section a unified vision of different {\it slow-light} based protocols. In this category, the ambassador is certainly the EIT scheme which has been particularly studied in the last decade with remarkable results in the quantum regime \cite{review_Ma_2017}. The linear dispersion associated with the EIT transparency window allows to define unambiguously a slow group velocity whose reduction to zero produces {\it stopped-light}. We have extended this concept to any retarded response that can be seen as a precursor for storage. This approach allows us to interpret the Raman scheme within the same framework. In that case, the group velocity cannot be defined {\it per se} but the dispersion profile still produces a retarded response that can be stored by shelving the excitation into a long lived spin state. The price to pay at the retrieval step is a significant pulse distortion even if the efficiency (input/output energy ratio) is quite satisfying. The pulse distortion at the retrieval is somehow a false problem. Distortions are more or less always present. Even in the more favorable EIT scheme, the pulse can be partially clipped because of a limited optical depth. It should be kept in mind that {quantum} repeater architectures {use} interference between outgoing photons \cite{simon2007, RevModPhys.83.33, bussieres2013prospective}. As soon as the different memories induce the same distortion, the retrieved outgoing {fields} can perfectly interfere. In that sense, the deformation can also be considered as an unitary transform between temporal modes without degrading the quantum information quality \cite{brecht2015photon, Thiel:17}.
The signal temporal deformation also raises the question of the waveform control through the storage step. We have used a simplistic model for the control field (on/off or $\pi$-pulses). A more sophisticated design of the control actually allows a build-in manipulation of the temporal and frequency modes of the stored qubit \cite{fisher2016frequency, conversion}. A quantum memory can be also considered a versatile light-matter interface with a enhanced panel of processing functions. Waveform shaping is not considered anymore as a detrimental experimental limitation but as new degree of freedom whose first benefit is the storage efficiency \cite{Novikova, zhou2012optimal} when specific optimization procedures are implemented \cite{GorshkovPRL, GorshkovII, nunnMultimode}. The optimization strategy by temporal shaping is beyond the scope of this chapter but would certainly {deserve} a review paper by itself.
The fast storage schemes and the EIT/Raman sequences that we analyzed in parallel in sections \ref{raman_stopped} and \ref{EIT_stopped} respectively, both rely on a Raman coupling field that control the storage and retrieval steps. The fast storage schemes depend on $\pi$-pulses and the EIT/Raman sequence on a control on/off switching. A three-level $\Lambda$-system seems to be necessary in that case. This is not rigorously true even if the $\Lambda$-structure is widely exploited for quantum storage. {\it Stopped-light} can indeed be obtained in two-level atoms by dynamically controlling the atomic properties \cite{simon_index, simon_dipole, chos}. Despite a lack of experimental demonstrations, these two-level alternative approaches conceptually extend the protocols away from the well-established atomic $\Lambda$-structure.
To close the loop with the previous section \ref{sec:2PE} on photon echo memories, we would like to discuss again the atomic frequency comb (AFC) protocol \cite{afc}. Despite its historical connection with the three-pulse photon echo sequence, {it has been argued that} the AFC falls in the {\it slow-light} memories \cite{afc_slow}. A judicious periodic shaping of the absorption profile, forming a comb, allows to produce an efficient echo. This latter can alternatively be interpreted as an undistorted retarded response using the terminology of section \ref{sec:SL}. This retarded part is a precursor that can be stored by shelving the excitation into spin states by a Raman $\pi$-pulse, thus definitely positioning the AFC in the fast storage schemes (section \ref{raman_stopped}).
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\section{Certifying the quantum nature of light storage protocols}\label{sec:certification}
The question that we address in this section is how to prove that a storage protocol operates in the quantum regime. The most natural answer is: { by demonstrating that the quantum nature of a light beam is preserved after storage}. There are, however, several ways for a {memory to output light beams that show} quantum features. It can simply be a light pulse, like a single photon, that cannot be described by a coherent state or a statistical mixture of coherent state \cite{glauber}.
Alternatively, a state can be qualified as being quantum when it leads to correlations between measurement results that cannot be reproduced by classical strategies based on pre-agreements and communications, as some entanglement states do. What is thus the difference between showing the capability of a given memory to store and retrieve single photons and entangled states ?
{Faithful} storage and retrieval of single photons {demonstrates} that the noise {generated by} the memory is low enough to preserve the photon statistics, even when {these statistics} cannot be reproduced by classical light. It does not show, however, that the memory preserves coherence. Furthermore, it does not prove that the memory cannot be reproduced by a classical strategy, that is, a protocol which would first {measure} the incoming photon and create another photon when requested.
{On the other hand}, the storage and retrieval of entangled states can be {implemented} to show that the memory outperforms any classical {measure}-and-prepare strategy. This is true provided that the fidelity of the storage protocol is high enough. For example, if a memory is characterized by storing one part of a two-qubit entangled state, the fidelity threshold is given by the fidelity of copies that would be created by a cloning machine taking one qubit and producing infinitely many copies. This is known to be one of the optimal {strategies} for determining an unknown qubit state \cite{Gisin1}.
Note that the fidelity reference can also be taken as the fidelity that would be obtained by a cloning protocol producing only one copy of the output state \cite{Scarani}. In this case, the goal is to {ensure} that the memory delivers the state with the highest possible fidelity, that is, if a copy exists, it cannot have a higher overlap with the input state. This {condition} is relevant whenever one wants to show the suitability of the memory for applications related to secure communications, {where third parties should not obtain information about the stored state} \cite{BB84}.
The suitability of a memory for secure communications can ultimately be certified {by} Bell tests \cite{Bell:1964kc}. In this case, the quality of the memory can be {estimated} without {assumptions} on the input state or on the measurements performed on the {retrieved} state. This {ensures} that the memory can be used in networks where secure communications can be realized over long distances with security guarantees holding independently of the details of the actual implementation.
We show in the following sections how these criteria can be tested in practice, describing separately benchmarks based on continuous and discrete variables. Various memory protocols are {used} as examples, including protocols such as {the two-pulse photon echo (2PE)} \cite{ruggiero} or the classical teleporter, which are known to be classical. In order to prove it, we first show how to compute the noise {inherent to} classical protocols by moving away from the semi-classical picture. While a fully quantized propagation model can be found in the literature \cite{GorshkovII} mirroring the semi-classical Schrödinger-Maxwell equations that we use in the previous sections, we present a toy model using an atomic chain to characterize memory protocols together with their noise (section \ref{toy_model}). Criteria are then derived first for continuous (section \ref{CV_criterion}) and then for {discrete} variables (section \ref{counting_criterion}).
}
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\subsection{Atomic chain quantum model}\label{toy_model}
The aim is to derive a simple quantum model allowing to characterize different storage protocols including the noise. Although quantum, the model is very simple and uses the basic tools of quantum optics.
\subsubsection{Jaynes-Cummings propagator}
We consider an electromagnetic field described by the bosonic operators $a$ and $a^\dag$ resonantly interacting with a single two-level atom (with levels $|g\rangle$ and $|e\rangle$) thought the Jaynes-Cummings Hamiltonian
\begin{equation}gin{equation}
\label{Jaynes_Cummings}
H_{\text{int}} = i \kappa (a^\dag \sigma_- - a \sigma_+).
\end{equation}
Here, $\sigma_\pm$ are atomic operators corresponding to the creation $\sigma_+ = |e\rangle\langle g| $ and annihilation $\sigma_- = |g\rangle\langle e|$ of an atomic excitation. The first term in \eqref{Jaynes_Cummings} is thus associated to the emission of a photon while the second term corresponds to its absorption. The corresponding propagator
\begin{equation}gin{equation}
U(\tau) = e^{\kappa \tau (a^\dag \sigma_- - a \sigma_+)} = \sum_{n \geq 0} \frac{(\kappa \tau)^n}{n!}(a^\dag \sigma_- - a \sigma_+)^n
\end{equation}
can be written as
\begin{equation}gin{align}
\nonumber
&& U(\tau) = \cos(\kappa \tau \sqrt{a^\dag a}) |g\rangle \langle g | - a \sin(\kappa \tau \sqrt{a^\dag a})/\sqrt{a^\dag a} |e\rangle\langle g| \\
\nonumber
&&\quad + \cos(\kappa \tau \sqrt{a a^\dag}) |e\rangle \langle e| + a^\dag \sin(\kappa \tau \sqrt{a a^\dag})/\sqrt{a a^\dag} |g\rangle\langle e|
\end{align}
by noting that
\begin{equation}gin{equation}
\nonumber
(a^\dag \sigma_- - a \sigma_+)^{2k} = (-1)^k \left(\left(a^\dag a \right)^k |g\rangle\langle g| + \left(a a^\dag \right)^k |e\rangle\langle e|\right),
\end{equation}
and
\begin{equation}gin{equation}
\nonumber
(a^\dag \sigma_- - a \sigma_+)^{2k+1} = (-1)^k \left(a^\dag \left(a a^\dag \right)^k \sigma_- - a \left(a^\dag a \right)^k \sigma_+\right).
\end{equation}
Hence, the following initial states read
\begin{equation}gin{align}
\nonumber
U(\tau) |g,0\rangle & \rightarrow |g,0\rangle,\\
\nonumber
U(\tau) |g,1\rangle& \rightarrow \cos(\kappa \tau) |g,1\rangle - \sin(\kappa \tau) |e,0\rangle,\\
\nonumber
U(\tau) |e,0\rangle &\rightarrow \cos(\kappa \tau) |e,0\rangle + \sin(\kappa \tau) |g,1\rangle.
\end{align}
\subsubsection{Absorption}
Let us now consider a collection of $N$ atoms, all prepared in the ground state $|g\rangle$ and each interacting with a single photon through the Jaynes-Cummings interaction. The state of the atoms associated with a successful absorption is given by
\begin{equation}gin{equation}
\nonumber
\rho_{\text{cond}} = \mathrm{Tr}_{\text{light}} \left[\ket{0}\bra{0} \otimes \id \-\ U_N \hdots U_1 |g\hdots g,1\rangle \langle g\hdots g,1 | U_1^\dag \hdots U_N^\dag\right]
\end{equation}
and takes the form $\rho_{\text{cond}} = |\Psi_{\text{cond}}\rangle\langle \Psi_{\text{cond}}|$ when applying explicitly the $N$ propagators, where
\begin{equation}gin{equation}
\nonumber
|\Psi_{\text{cond}}\rangle =c^{N-1}s |g\hdots g e\rangle + c^{N-2} s |g\hdots e g \rangle + \hdots + s |e g \hdots g\rangle.
\end{equation}
Note that we have introduced the shorthands $c=\cos(\kappa \tau)$ and $s=\sin(\kappa \tau).$ The normalization of $\Psi_{\text{cond}},$ that is $1-\cos^{2N}(\kappa \tau),$ gives the probability of a successful absorption. For a small absorption amplitude per atom $\kappa \tau = \sqrt{d /N} \ll 1$ where $d=\alpha L$ is the total optical depth of the atomic chain and a large atom number, we have
\begin{equation}gin{equation}
\lim_{N\rightarrow \infty} \cos^{2N}(\kappa \tau) \approx \lim_{N\rightarrow \infty} \left(1-\frac{d}{2N}\right)^{2N} \rightarrow e^{-d}
\end{equation}
which corresponds to the Bouguer-Beer-Lambert absorption law (eq.\ref{bouguer}). Similarly, the absorption probability $1- \cos^{2N}(\kappa \tau)$ tends to $1-e^{-d}.$
\subsubsection{Storage and retrieval probability}
The overall efficiency including the storage and retrieval probabilities is obtained by calculating
\begin{equation}gin{equation}
\nonumber
|\langle g\hdots g, 1| U_N \hdots U_1 |\Psi_{\text{cond}}\rangle|^2.
\end{equation}
Note that we here consider a forward emission in which the retrieved photon is emitted in the same direction that the input photon. We obtain
\begin{equation}gin{align}
\nonumber
&&|\langle g\hdots g, 1| U_N \hdots U_1 |\Psi_{\text{cond}}\rangle|^2 = N^2 s^4 c^{2N-2} \\
&& \approx d^2 \left(1-\frac{d}{N}\right)^{2N-2} \rightarrow d^2 e^{-d} \-\ \text{when} \-\ N \rightarrow \infty
\end{align}
and thus retrieve the semi-classical forward efficiency eq.\eqref{eta_crib}.
For a backward emission, we obtain
\begin{equation}gin{equation}
|\langle g\hdots g, 1| U_1 \hdots U_N |\Psi_{\text{cond}}\rangle|^2 \rightarrow (1-e^{-d})^2 \-\ \text{when} \-\ N \rightarrow \infty
\end{equation}
corresponding to the semi-classical backward efficiency eq.\eqref{eta_crib_back}.
\subsubsection{Amplification through an inverted atomic ensemble} This simple model allows {us} to compute the expected noise of protocols for which the excited states are significantly populated when the stored excitation is released as in the two-pulse photon echo (2PE) protocol described in section \ref{2PE}. Consider first the case where all the atoms are in $|e\rangle$ and the field is in the vacuum state $|0\rangle.$ The mean photon number after an interaction time $\tau$ is given by
\begin{equation}gin{equation}
\nonumber
\langle e \hdots e,0 | U_N^\dag(\tau) \hdots U_1^\dag(\tau) a^\dag a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle
\end{equation}
which can be seen as the square of the norm of $a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle.$ In the regime where the population remains essentially in the excited state, the atomic operators $\sigma_\pm$ verifies
\begin{equation}gin{equation}
[\sigma_+, \sigma_-] = |e\rangle\langle e| - |g\rangle \langle g| \approx 1.
\end{equation}
In this case, the Hamiltonian \eqref{Jaynes_Cummings} is a squeezing operator between two bosonic modes and the formula $e^B A e^{-B} = \sum_{n\geq 0} \frac{1}{n!} \underbrace{[B,\hdots [B, A]\hdots ]}_{\text{n times}}$ can be used to prove that
\begin{equation}gin{equation}
a U_1 = U_1 \left(\cosh(\kappa \tau) a + \sinh(\kappa \tau) \sigma_-^{(1)}\right).
\end{equation}
Commuting $a$ with $U_2 \hdots U_N,$ we obtain
\begin{equation}gin{align}
\label{transform_field_s}
&a U_1 \hdots U_N = U_1 \hdots U_N \Big(\cosh(\kappa \tau)^N a \\
\nonumber
&+ \cosh(\kappa \tau)^{N-1} \sinh(\kappa \tau) \sigma_-^{(N)} + \hdots +\sinh(\kappa \tau) \sigma_-^{(1)} \Big)
\end{align}
where $\sigma_-^{(i)}$ is the atomic operator $\sigma_-$ for the $i^{th}$ atom. This leads to
\begin{equation}gin{align}
\nonumber
& || a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle ||^2 \\
\nonumber
&= \sinh(\kappa \tau)^2 \sum_{j=1}^N \cosh(\kappa \tau)^{2j-2} = \cosh(\kappa \tau)^{2N}-1.
\end{align}
Using $\kappa \tau = \sqrt{d/N} \ll 1$ and taking the limit of large $N$, the mean photon number is
\begin{equation}gin{equation}
e^d-1.
\end{equation}
This corresponds to the number of photons emitted in a single mode by an inverted ensemble \cite{RASE, Sekatski}.
More generally, eq.\eqref{transform_field_s} shows that in the regime where the atoms are mainly in the excited state, the atomic ensemble operates as a classical amplifier, the gain $G$ depending exponentially on the optical depth via $G=e^{d}.$ Such an amplifier transforms the field operators according to
\begin{equation}gin{align}
\label{a_out}
& \bar U^\dag a \bar U = \sqrt{G} a + \sqrt{G-1} \-\ \sigma_c^\dag \\
\label{adag_out}
& \bar U^\dag a^\dag \bar U = \sqrt{G} a^\dag + \sqrt{G-1} \-\ \sigma_c
\end{align}
where $\bar U^\dag a \bar U = U_N^\dag (\tau) \hdots U_1^\dag (\tau) a U_1(\tau) \hdots U_N(\tau).$
The bosonic operators $\sigma_c$ and $\sigma_c^\dag$ annihilates and creates collectively atoms in the ground state
$$
\sigma_c^\dag = \frac{1}{\sqrt{M}}(\cosh(\kappa \tau)^{N-1} \sinh(\kappa \tau) \sigma_-^{(N)} + \hdots +\sinh(\kappa \tau) \sigma_-^{(1)})
$$
with the normalization coefficient $M=\cosh(\kappa \tau)^{2N}-1.$\\
Equations \eqref{a_out}\&\eqref{adag_out} allow one to derive expectations values for the field when the atoms are mostly excited. Let us consider for example a 2PE where the first pulse is a single photon Fock state and the second pulse is a $\pi$-pulse. We can compute the response of a non-photon number resolving detector with dectection efficiency $\eta$ at the echo time. If the photon has been successfully absorbed, at the echo time, the atoms are well described by a single excitation $|1\rangle$ in the collective mode $\sigma_c$ and the field $a$ is in the vacuum state. The probability that the photon detector clicks is given by
\begin{equation}gin{align}
\nonumber
\langle 01| \bar U^\dag \big(1-(1-\eta)^{a^\dag a}\big) \bar U |01\rangle & = 1 - \langle 01| \bar U^\dag :e^{-\eta a^\dag a} : \bar U |01\rangle \\
\nonumber
& = 1- \langle 01| \bar U^\dag (1 -\eta aa^\dag + \frac{\eta^2}{2} a^\dag a ^\dag a a - \hdots) \bar U |01\rangle.
\end{align}
Using eqs.\eqref{a_out}\&\eqref{adag_out}, we easily show that
$$ \langle \bar U^\dag \underbrace{a^\dag \hdots a ^\dag}_{\text{k times}} \underbrace{a \hdots a}_{\text{k times}} \bar U\rangle = (k+1)! (G-1)^k.$$
Therefore
\begin{equation}gin{align}
\nonumber
\langle 01| \bar U^\dag \big(1-(1-\eta)^{a^\dag a} \big) \bar U |01\rangle & = 1- \sum_{k\geq 0} (-1)^k \eta^k (G-1)^k (k+1)\\
\label{clic_2pulse}
& = 1- \frac{1}{\big(1+\eta(G-1)\big)^2}.
\end{align}
This formula shows that no click is obtained when the detection efficiency is null while the detectors clicks with unit probability as long as $\eta G \gg 1.$
\subsubsection{Beamsplitter interaction in a non-inverted ensemble}
In the regime where the atoms are and remain essentially in the ground state, the atomic operators $\sigma_\pm$ verifies
\begin{equation}gin{equation}
[\sigma_+, \sigma_-] = |e\rangle\langle e| - |g\rangle \langle g| \approx -1.
\end{equation}
As in the previous paragraph, the formula $e^B A e^{-B} = \sum_{n\geq 0} \frac{1}{n!} [B, A]^n$ can thus be used to prove that in this regime
\begin{equation}gin{equation}
a U_N = U_N \left(\cos(\kappa \tau) a + \sin(\kappa \tau) \sigma_-^{(N)}\right).
\end{equation}
We thus have
\begin{equation}gin{align}
\label{transform_field}
&a U_N \hdots U_1 = U_N \hdots U_1 \Big(\cos(\kappa \tau)^N a \\
\nonumber
&+ \cos(\kappa \tau)^{N-1} \sin(\kappa \tau) \sigma_-^{(1)} + \hdots + \sin(\kappa \tau) \sigma_-^{(N)} \Big).
\end{align}
By introducing the collective operator
\begin{equation}gin{equation}
\bar{\sigma}_c = \frac{1}{\sqrt{\overline M}}(\cos(\kappa \tau)^{N-1} \sin(\kappa \tau) \sigma_-^{(N)} + \hdots +\sin(\kappa \tau) \sigma_-^{(1)})
\end{equation}
with $\overline M = 1-\cos(\kappa \tau)^{2N},$ the atom-light interaction can be seen as a standard beamsplitter-type interaction
\begin{equation}gin{align}
\label{a2}
& \bar U^\dag a \bar U = \sqrt{e^{-d}} a + \sqrt{1-e^{-d}} \-\ \bar{\sigma}_c \\
\label{adag2}
& \bar U^\dag a^\dag \bar U = \sqrt{e^{-d}} a^\dag + \sqrt{1-e^{-d}} \-\ \bar{\sigma}_c^\dag.
\end{align}
The formulas derived from this simple quantum model will be helpful to characterize the quantum nature of different storage protocols as we will see now.
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\subsection{Continuous variable criterion}\label{CV_criterion}
Here, we study the propagation and read-out of a pulse with quantum noise through different memories and {review} a criterion to evaluate if {the output state is the best cloned copy of the input, that is, to guarantee that no better copy of the input state is available}. We analyze generic storage protocols in a continuous variable perspective to estimate the amount of noise and loss that can be tolerated {to fulfill this criterion}.
\subsubsection{The stored quantum states}
A quantum memory should be able to store and retrieve any state {while preserving its quantum features}. The state can be a classical state but its quantum statistics should be preserved. In continuous variable quantum information, the variables of interest are the field quadratures, defined as
\begin{equation}gin{equation}X^+=a+a^\dagger\end{equation} and
\begin{equation}gin{equation}X^-=-i(a-a^\dagger)\end{equation}
where $a$ and $a^\dagger$ are the creation and annihilation operators of the field, as in the previous section. As they satisfy the canonical commutation relations $[a,a^\dagger]=1$, it follows that $[X^+,X^-]=2i$ and that
\begin{equation}gin{equation}n=\frac{1}{4}\big[(X^+)^2+(X^-)^2\big]-\frac{1}{2}\end{equation}
where $n=a^\dagger a$ is the photon number operator.
The signal at the output of a quantum memory can be decomposed into a classical amplitude $\alpha$ and a fluctuating noise term $\delta\hat{X}^\pm$.
Formally, for a gaussian state, we write the amplitude and phase quadratures of the field as
\begin{equation}gin{equation}\hat{X}^\pm=\alpha^\pm+\delta\hat{X}^\pm \end{equation}
To avoid writing the propagators when describing the field at the output of the memory, we now introduce the subscript $_{\rm {out}}$ defined as $a_{\rm {out}}= \overline{U}^\dagger a\overline{U}$ for example. Similarly, the subscript $_{\rm {in}}$ is used to describe the input of the memory.
The measured output signal is generally the power spectral density, given by the Fourier transform of the autocorrelation function.
It reads as
\begin{equation}gin{equation}S_{\rm out}^\pm=\langle (\hat{X}_{\rm out}^\pm)^2 \rangle \end{equation}
and the noise as
\begin{equation}gin{equation}V_{\rm out}^\pm=\langle \delta (\hat{X}_{\rm out}^\pm)^2 \rangle \end{equation}
We thus obtain
\begin{equation}gin{equation}S_{\rm out}^\pm=(\alpha_{\rm out}^\pm)^2+V_{\rm out}^\pm.\end{equation}
We will estimate $\alpha_{\rm out}^\pm$ and $V_{\rm out}^\pm$ at the output of the optical memories and identify the values that enable entering the quantum memory regime.
\subsubsection{Quantum memory criterion}
Generally, optical memories are benchmarked against quantum information criteria. In particular, the performance of a given quantum memory can be {evaluated similarly to a quantum teleportation scheme} {by quantifying the quality of the output state with respect to the input.}
Figure \ref{carct} shows the schematics of the quantum memory benchmark. The optimal classical measure and prepare strategy for optical memory consists in measuring the input state jointly on two conjugate quadratures using two homodyne schemes \cite{Hammerer}. The measured information is stored before fed-forward onto an independent beam. In this classical scheme, the storage time can be arbitrarily long without additional degradation. However, two conjugate observables cannot be simultaneously measured and stored without paying a quantum of duty. Moreover, the encoding of information onto an independent beam will also introduce another quantum of noise. In total, the entire process will incur two additional quanta of noise onto the output optical state \cite{HetetPRA}.
Characterizing quantum memory using the state-dependent fidelity as a measure can be complicated for exotic mixed states. Alternatively, we use the signal transfer coefficients $T$ and the input-output conditional variances $V_{cv}$ to establish the efficiency of a process \cite{Grangier, Ralph}. The conditional variances and signal transfer coefficients are defined as
\begin{equation}gin{equation} V_{cv}^{\pm}=V_{\rm out}^{\pm} -\frac{ |\langle X^\pm_{\rm in} X^\pm_{\rm out} \rangle|^2}{V^\pm_{\rm in}} \end{equation}
and
\begin{equation}gin{equation} T^\pm = \frac{R^\pm_{\rm out}}{R^\pm_{\rm in}} \end{equation}
where $R^\pm_{\rm out/in}$ is defined as
\begin{equation}gin{equation} R^\pm_{\rm out/in}=\frac{4(\alpha^\pm_{\rm out/in})^2}{V^\pm_{\rm out/in}}. \end{equation}
We now define two parameters that take into account the performances of the system on both conjugate observables as
\begin{equation}gin{equation} V=\sqrt{V_{cv}^{+}V_{cv}^{-}} \end{equation}
and \begin{equation}gin{equation} T=T^++T^- \end{equation}
It can be shown that a classical memory {based on the measure and prepare scheme described before} cannot overcome the $T>1$ or $V<1$ limits \cite{Ralph}. With a pair of entangled beams, it is possible to have an output state with $V<1$ or $T>1$, hence demonstrating that the memory outperforms the optimal measure and prepare strategy. In case where the output state satisfies both $V<1$ and $T>1,$ the output is the best possible cloned copy of the input state \cite{Grosshans}.
A perfect quantum memory would satisfy both $T=2$ and $V=0$.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.3}{\includegraphics{Fig_CVcriterion.eps}}}
\caption{General scheme for characterizing an optical memory. A pair of EPR entangled beams with a mean signal amplitude is prepared. One of these beams is injected into, stored, and readout from the optical quantum memory (QM) while the other is being propagated in free space. A joint measurement with appropriate delay is then used to measure the quantum correlations between the quadratures of the two beams.
}\label{carct}
\end{figure}
\subsubsection{{\it Slow-light} memory}\label{CV_SL}
\begin{equation}gin{figure}[hb!]
\centerline{\scalebox{0.9}{\includegraphics{gain.eps}}}
\caption{Field propagating in a medium with gain $\alpha$ and loss $\begin{equation}ta$.
}\label{Setup}
\end{figure}
We now present a general theory for amplification and attenuation of a traveling wave and use the TV diagram to quantify the amount of excess noise that is tolerated.
This theory is well adapted to {\it slow-light} memories \cite{HetetPRA} but can be carried over to other memories, like Raman (section \ref{Raman}) or CRIB (section \ref{CRIB}) memories. Gain can indeed be present if, for instance, population has been transferred to other states during the mapping and read-out stages.
As discussed in the previous section, and in particular in eqs.\eqref{a_out}\&\eqref{adag_out}, the output of an ideal linear amplifier with a gain factor $G>1$, relates to the input field {\it via} this relation :
\begin{equation}gin{equation} a_{\rm out}=\sqrt{G} a_{\rm in}+\sqrt{G-1}\sigma_c^\dagger \end{equation}
where $\sigma_c^{\dagger}$ is a bosonic operator in the vacuum state. The power
spectrum at the output of an ideal phase-insensitive amplifier
is then given by \begin{equation}gin{equation}S_{\rm out}=GS_{\rm in} +G-1 \end{equation} where $S_{\rm in}$ is input spectrum.
By concatenating $m$ amplifying and attenuating infinitesimal slices with linear amplification $1+\alpha \delta z$ and attenuation
$1-\begin{equation}ta \delta z$ where $\delta z=z/m$, as represented in fig.\ref{Setup}, we will calculate the noise properties of the field.
The power spectrum of the field at a slice $m$ is
\begin{equation}gin{equation}
S_m=(1+\frac{(\alpha-\begin{equation}ta)z}{m})^m (S_{in}-1) +1+2\alpha\sum (1+\frac{(\alpha-\begin{equation}ta)z}{m})^{m-j}
\end{equation}
\\
In the infinitesimal slice width limit, we obtain
\begin{equation}gin{equation} S_{\rm out}=\eta S_{\rm in} +(1-\eta)(1+N_f) \end{equation}
where $N_f=2\alpha/(\begin{equation}ta-\alpha)$ and $\eta=\exp{((-\begin{equation}ta+\alpha)L)}$ where $L$ is the length of the medium.
Using standard memory protocols, one can find a relationship between $\alpha$, $\begin{equation}ta$ and the memory parameters.
One can then show that
\begin{equation}gin{equation} V=1-\eta+V_{\rm noise} \end{equation}
and \begin{equation}gin{equation} T=2\eta/(1+V_{\rm noise}),\end{equation} where $V_{\rm noise}=1+(1-\eta) N_f$.
Figure \ref{TV} shows a TV diagram for a memory with varying loss (arrows) and three different gain values.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.7}{\includegraphics{TV.eps}}}
\caption{TV diagram for a CRIB memory with varying gain and loss. The dashed line shows the evolution of the standard 2PE memory performance as a function of optical depth.
}\label{TV}
\end{figure}
If there is no mean intensity of the field at the input, the output field is simply the memory output noise. It reads as
\begin{equation}gin{equation} S_{\rm out}=\eta +(1-\eta) (1+N_f) \end{equation}
If we further assume that all the atoms are in the excited state, that is the atomic
medium operates as an amplifier ($\begin{equation}ta$=0), $N_f=-2$, now we obtain
\begin{equation}gin{equation} S_{\rm out}=1-2(1-\eta)=2\eta-1 \end{equation}
Assuming that the noise is the same on both quadratures, and the relation between the mean number of photons and the field quadratures leads to
\begin{equation}gin{equation} \langle n \rangle=\frac{1}{2}\big[\langle X ^2 \rangle-1\big]
= \frac{1}{2}\big[S_{\rm out}-1\big]=\eta-1
\end{equation}
The mean number of photon is thus
\begin{equation}gin{equation} \label{mean_photon_number}
e^{d}-1
\end{equation}
where $d=\alpha L$ in the optical depth, as was found in the previous section.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.55}{\includegraphics{2PE_CV.eps}}}
\caption{Beam splitter description of photon echo memories.}\label{BS}
\end{figure}
\subsubsection{Photon echo memories}
Standard photon echo protocols that use long lived excited state transitions are generally not immune to noise. If the emission takes place while population remains in the excited state, gain will be present so the memory will not enter the quantum regime.
\paragraph{Controlled reversible inhomogeneous broadening}
The CRIB scheme can be modeled using arrays of beam-splitters. In its most efficient form, namely using the gradient echo memory scheme (GEM) \cite{hetet2008electro} (sometimes called longitudinal CRIB) or using (transverse-) CRIB with a backward write pulse
(section \ref{CRIB}), the write and read stages can be seen as two beam-splitters with a reflectivity that depends on the optical depth \cite{LongdellAnalytic}, as depicted in fig.\ref{BS}.
Let us note that without a backward pulse, more beam-splitters are needed to describe the output field of the CRIB memory \cite{LongdellAnalytic}.
In these scheme, the population remains mainly in the ground state so that gain, and thus noise, will be absent.
For the write stage, the transmitted pulse field intensity is attenuated according to the Bouguer-Beer-Lambert law by $e^{-d}$.
In terms of quadratures, including the "vacuum port" modeling atomic fluctuations, we deduce from eqs.\eqref{a2}\&\eqref{adag2} the expressions for the light and spin quadratures at the two output ports as defined in fig.\ref{BS}
\begin{equation}gin{equation} X_t=\sqrt{e^{-d}} X_{\rm in} - \sqrt{1-e^{-d}}X_{c1} \end{equation}
and
\begin{equation}gin{equation} X_s=\sqrt{1-e^{-d}} X_{\rm in} + \sqrt{e^{-d}}X_{c1} \end{equation}
The vacuum contribution ensures preservation of the commutation relations of the field and atomic operators.
In the case of CRIB with backward propagation or GEM (forward), the beam-splitter reflectivity is "inverted" and the output field can be written simply as
\begin{equation}gin{equation} X_{\rm out}=\sqrt{1-e^{-d}} X_{s} + \sqrt{e^{-d}}X_{c2}.\end{equation}
In the absence of signal, the input is in the vacuum state $V_{\rm in}=1$, so that $V_{\rm out}=1$.
We thus find a conditional variance
\begin{equation}gin{equation} V_{cv}^{\pm}=1-(1-e^{-d})^2 \end{equation}
and transfer coefficient.
\begin{equation}gin{equation} T=2 (1-e^{-d})^2.\end{equation}
So for the present case of CRIB and in the limit of large optical depth, $T\rightarrow 2$ and $V\rightarrow 0$, the CRIB memory is a quantum memory, as represented by the green area in fig.\ref{TV}.
\paragraph{Standard two-pulse photon echo}
Let us now consider the 2PE memory described in section \ref{2PE}. The difference between the 2PE memory and the CRIB is that the atoms are in the excited state during the read-out retrieval stage. This inversion implies that the input light will be amplified, which will invariably add noise.
Considering again the two-beam-splitter approach depicted in fig.\ref{BS}, the writing stage is the same as CRIB, so a fraction $\sqrt{\eta_R}=\sqrt{1-e^{-d}}$ of the field is written in the memory.
The output field however is amplified by a quantity $\sqrt{\eta_W}=\sqrt{e^d-1}$ as discussed in the {\it slow-light} section \ref{CV_SL}.
In total, the transmission thus reads \begin{equation}gin{equation}\eta_R \eta_W=(1-e^{-d})(e^d-1)=4~{\rm sinh}^2(d/2)\end{equation}We here retrieve the semi-classical 2PE efficiency eq.\eqref{etaPi}.
In terms of the field quadratures, we have
\begin{equation}gin{equation} X_{\rm s}=\sqrt{1-e^{-d}} X_{\rm in}+\sqrt{e^{-d}}X_{\rm c1}\end{equation}
and
\begin{equation}gin{equation} X_{\rm out}=\sqrt{e^{d}-1} X_{\rm s}+\sqrt{e^{d}}X_{\rm c2}\end{equation}
just like for a linear amplifier (eqs.\ref{a2}\&\ref{adag2}).
The product of the conditional variances is thus
\begin{equation}gin{equation}V=1-e^{-d}+e^d\end{equation}
and the sum of the two signal-to-noise transfer coefficients is
\begin{equation}gin{equation}T=\frac{4 {\rm sinh}(d/2)^2}{2 e^d-1}\end{equation}
These two quantities are plotted in fig.\ref{TV} (dotted line), showing the 2PE memory does not enter the quantum regime for any optical depth.
We have checked that a better performance (a lower V and larger T) can be obtained if the optical depth is lowered during the writing stage but the memory would still operate in the classical domain.
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\subsection{Photon counting criteria}\label{counting_criterion}
We now present various criteria for certifying the quantum nature of storage protocols based on photon counting, including the autocorrelation measurement, the Cauchy-Schwarz criterion and the Bell test.
\subsubsection{Autocorrelation measurement}
Let us consider a single-mode of the electromagnetic field with bosonic operators $a$ and $a^\dag$ and described by the state $\rho_a.$ This state is said classical if it can be represented as a mixture of coherent states $|\alpha\rangle$, that is, one can find a quasi-probability distribution $P(\alpha) \geq 0$ such that
\begin{equation}gin{equation}
\rho_a=\int d^2 \alpha P(\alpha) |\alpha\rangle\langle \alpha|.
\end{equation}
The autocorrelation of this field defined as
\begin{equation}gin{equation}
g_a^{(2)}=\frac{\langle a^{\dag 2} a^2 \rangle}{\langle a^\dag a \rangle^2}
\end{equation}
is at least equal to 1 \cite{loudon2000quantum}. Conversely, if the result of an autocorrelation measurement is smaller than 1, one can conclude that the measured state is non-classical. A single photon Fock state for example, is a non-classical state because its autocorrelation is 0. A simple way to certify the quantum nature of a given memory is thus to store a single photon and to check that the result of an autocorrelation measurement after retrieval is smaller than 1. This shows that the memory preserves the non-classical feature of light.
Note that in practice, non photon-number resolving detectors can be used to certify the non-classical nature of a single-mode field: it is sufficient to put two of these detectors after a 50/50 beamsplitter and to check that the probability of a twofold coincidence is smaller than the product of probabilities of singles \cite{sekatski2012detector}. Let us thus consider the experiment represented in fig.\ref{Fig1} where a source produces a single photon that is subsequently stored in a memory. The photon is then released and an autocorrelation measured with non photon-number resolving detectors ($d_a$ and $\bar{d}_a$) with efficiency $\eta_d$ each. Let $\eta_m$ be the efficiency of the memory and $p_{{\text{dc}}}$ the probability to get a dark count, that is a click on one detector when the photon source is switched off. Obviously, $\eta_m$ can include the non-unit efficiency of the source and the loss from the source to the memory. $\eta_d$ also accounts for the loss between the 50/50 beamsplitter and each detector. $p_{{\text{dc}}}$ includes the detector dark counts and various sources of noise operating independently on each detector. We assume that $p_{{\text{dc}}}$ is the same for both detectors ($d_a$ and $\bar{d}_a$). To obtain twofold coincidences smaller than the product of singles, these parameters has to fulfill the following inequality (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{equation}
\label{autocorrelation}
g_a^{(2)}=\frac{1-2(1-p_{\text{dc}})(1-\eta_d\eta_m/2)+(1-p_{\text{dc}})^2(1-\eta_d\eta_m)}{\left(1-(1-p_{\text{dc}})(1-\eta_d\eta_m/2)\right)^2} < 1.
\end{equation}
Note that in the absence of noise ($p_{\text{dc}}=0$), this ratio is zero independently of the efficiency. In other words, for an ideal implementation of a memory protocol without noise, there is no constraint on the memory efficiency to prove that it can preserve the result of an autocorrelation measurement performed on a single photon. For unit efficiencies $\eta_d\eta_m =1,$ the ratio eq.\eqref{autocorrelation} tends to $1-\epsilon^2/4$ for $p_{\text{dc}} \approx 1-\epsilon.$ For low efficiencies $\eta_d\eta_m \ll 1,$ the inequality \eqref{autocorrelation} is fulfilled as long as ${p_\text{dc}} \leq 3\eta_d\eta_m.$
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig1_counts.eps}
\caption{Certifying the quantum nature of a memory by checking that it can preserve the non-classical property of a single-mode field. A single photon Fock state is stored and an autocorrelation measurement is performed on the retrieved photon.}
\label{Fig1}
\end{figure}
It is worth mentioning that the here proposed criterion does not allow to conclude that the memory outperforms classical strategies for storage as a device that would throw the photon emitted by the source and create a photon afterward would lead to a zero autocorrelation measurement. However, under the assumption that memory under test indeed operates as a storage/retrieval protocol, this criterion shows that this memory is in the quantum regime, in the sense that it preserves the non-classical nature of a single mode field.
It is interesting to compute the result of an autocorrelation measurement that would be obtained by storing a single photon in an atomic ensemble using the 2PE technique. In the ideal scenario where there is no loss before and inside the memory and the photon absorption is successful, we find for the autocorrelation
\begin{equation}gin{equation}
\label{g22pe}
\frac{1-\frac{2}{(1+\frac{\eta_d}{2}(e^d-1))^2} + \frac{1}{(1+\eta_d(e^d-1))^2}}{\left(1-\frac{1}{(1+\frac{\eta_d}{2}(e^d-1))^2}\right)^2}
\end{equation}
which tends to $3/2$ for small optical depth $d\ll 1$ and to 1 when $\eta/2 (e^{d}-1) \geq 1.$ Even in an ideal scenario, we conclude that a storage technique based on a two-pulse photon echo does not preserve the non-classical nature of a single photon.
\subsubsection{Cauchy-Schwarz criterion}
The Cauchy-Schwarz parameter R can be used to reveal non-classical correlations between two fields \cite{clauser1974experimental}. Consider two single-mode fields and their respective bosonic operators $a,$ $a^\dag$ and $b,$ $b^\dag.$ We say that these fields are classically correlated if their state $\rho_{ab}$ can be written as a mixture of coherent states $|\alpha\rangle$, $|\begin{equation}ta\rangle,$ that is, there exists a non negative function $P(\alpha, \begin{equation}ta)$ such that
\begin{equation}gin{equation}
\rho_{ab}=\int d^2\alpha d^2\begin{equation}ta P(\alpha, \begin{equation}ta) |\alpha, \begin{equation}ta\rangle\langle \alpha, \begin{equation}ta|.
\end{equation}
The Cauchy-Schwarz parameter defined as
\begin{equation}gin{equation}
R=\frac{\langle a^\dag b^\dag b a \rangle}{ \langle a^{\dag 2} a^2 \rangle \langle b^{\dag 2} b^2 \rangle}
\end{equation}
is at most equal to 1 when calculated on classically-correlated states. $R > 1$ is a witness of non-classical correlations.
As for the autocorrelation measurement, the Cauchy-Schwarz parameter can be measured with non photon-number resolving detectors \cite{sekatski2012detector}, see fig.\ref{Fig2}. It is sufficient to take the ratio between the square of twofold coincidences between detectors $d_a$\&$d_b$ and the product of coincidences between $d_a$\&$\bar{d}_a$ and $d_b$\&$\bar{d}_b.$ Let us consider the experiment shown in fig.\ref{Fig2} with a source producing two-mode vacuum squeezed states, that is
\begin{equation}gin{equation}
(1-p)^{\frac{1}{2}} e^{\sqrt{p} a^\dag b^\dag} |00\rangle.
\end{equation}
Further consider the storage and release of the mode $a$ into a memory with efficiency $\eta_m.$ Let $\eta_{d}^a$ ($\eta_{d}^b$) be the efficiency of detectors $d_a$ and $\bar{d}_a$ ($d_b$ and $\bar{d}_b$) and $p_{\text{dc}}^a$ ($p_{\text{dc}}^b$) the probability to get a click on the detector $d_a$ or $\bar d_a$ ($d_b$ or $\bar d_b$) when the source is switched off (dark counts). As before, the memory efficiency includes the loss from the source to the memory. The efficiency of detectors $d_a$ and $\bar{d}_a$ includes the loss from the beamsplitter to the detector while the efficiency of the detectors $d_b$ and $\bar{d}_b$ includes the loss from the source to the detector (without the transmission of the beamsplitter). $p_{\text{dc}}^a$ which we assume to be the same for the two detector $d_a$ and $\bar{d}_a,$ includes various source of noise that can be modeled as detector dark counts. In this scenario, the Cauchy-Schwarz parameter is given by (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{align}
\label{Cauchy_Schwarz}
R=&\Bigg[1-\frac{(1-p_{\text{dc}}^a)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})}-\frac{(1-p_{\text{dc}}^b)(1-p)}{1-p(1-\frac{\eta_d^b}{2})}
+\frac{(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})(1-\frac{\eta_d^b}{2})}\Bigg]^2/\nonumber \\
\nonumber
& \Bigg[\left(1-2\frac{\left(1-p_{\text{dc}}^{a}\right)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})}+\frac{\left(1-p_{\text{dc}}^{a}\right)^2(1-p)}{1-p(1-\eta_d^a\eta_m)}\right)\times\\
&\left(1-2\frac{\left(1-p_{\text{dc}}^{b}\right)(1-p)}{1-p(1-\frac{\eta_d^b}{2})}+\frac{\left(1-p_{\text{dc}}^{b}\right)^2(1-p)}{1-p(1-\eta_d^b)}\right)\Bigg]
\end{align}
and has to be larger than $1$ to certify that the tested memory preserves non-classical correlations. In the ideal case with unit efficiencies and no dark count, the Cauchy-Schwarz parameter tends to $\frac{1}{4}(1+\frac{1}{p})^2$ for $p \ll 1$. Note that $p$ can be written as a function of the mean photon-number emitted in one mode ($a$ or $b$) as $p=n/(n+1).$
The Cauchy-Schwarz criterion leads to similar conclusions than the autocorrelation measurement. If the memory under test is a device that throws the incoming field away and produces a single photon at a later time, the Cauchy-Schwarz parameter would tend to infinity, independently of the state of mode $b$. However, assuming that the tested memory indeed operates as a storage/retrieval protocol, the Cauchy-Schwarz criterion allows to conclude that the memory preserves non-classical correlations between two fields.
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Fig2_counts.eps}
\caption{Setup to certify the quantum nature of a memory by checking that it can preserve the non-classical correlations between two fields. A photon pair source is used to produce two-mode squeezed vacuum states. One of the two modes (mode $a$) is stored in a memory and the Cauchy-Schwarz parameter is measured between the mode $a$ after retrieval and the mode $b.$}
\label{Fig2}
\end{figure}
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\subsubsection{Bell test}
The two criteria presented previously do not test the capability of a memory to preserve the coherence. This can be done by storing a part of an entangled state and by checking that the entanglement is preserved using, for example, a Bell test. A possible realization would use a photon pair source emitting entangled photon pairs, for example, in polarization. The spatial mode $a$ is stored in a memory and subsequently released. Measurements are finally performed combining wave-plates, polarizing beamsplitters and one detector on each side. The twofold coincidences are recorded. Two interference patterns are obtained, one by rotating the analyzer on the left side, the other one by rotating the analyzer on the right side. If the mean visibility of this interference patterns is larger than $1/3,$ one can conclude about the presence of entanglement under the assumption that the state is a mixture between the singlet state and white noise. As the memory operates as a local operation, which cannot increase entanglement, a high interference visibility witnesses the presence of entanglement between the photon in $b$ and the excitation stored in the memory. Note that there is no need to close the detection and locality loopholes here as the Bell test is used as an entanglement witness, not as non-locality Bell test.
Let us consider the experimental realization shown in fig.\ref{Fig3} with a source based on spontaneous parametric down conversion, that is, photon pairs described by
\begin{equation}gin{equation}
|\psi^-_{a_h a_v b_h b_v}\rangle = (1-p) e^{\sqrt{p} (a_h^\dag b_v^\dag-a_v^\dag b_h^\dag)} |00\rangle.
\end{equation}
Let $\eta_a$ and $\eta_b$ be the detector efficiency on side $a$ and $b$ respectively and $p_{\text{dc}}^a$ and $p_{\text{dc}}^b$ the corresponding noise. As before, the memory efficiency is labeled $\eta_m.$ The visibility of the interference $V$ is given by (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{align}
\label{Bell}
V=\Bigg[\frac{(1-p)(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{1-p(1-\eta_a)(1-\eta_b)} -
\frac{(1-p)^2(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{(1-p(1-\eta_a))(1-p(1-\eta_b))}\Bigg]/\nonumber \\
\Bigg[ 2-2\frac{(1-p_{\text{dc}}^a) (1-p)}{1-p(1-\eta_a)}-2\frac{(1-p_{\text{dc}}^b) (1-p)}{1-p(1-\eta_b)}+
\frac{(1-p)(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{1-p(1-\eta_a)(1-\eta_b)} +\nonumber \\
\frac{(1-p)^2(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{(1-p(1-\eta_a))(1-p(1-\eta_b))}
\Bigg].
\end{align}
As before, $p$ can be written as a function of the mean photon number in one mode ($a_h,$ $a_v,$ $b_h$ or $b_v$) as $p=n/(n+1).$
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Fig3_counts.eps}
\caption{Setup to certify the quantum nature of a memory by checking that it can preserve entanglement between two fields. A photon pair source is used to produce entangled photon pairs in polarization. Two of the four modes (mode $a_h$ and $a_v$) are stored in a memory and a Bell inequality violation can be inferred from the visibility of the interference that is obtained by recording the twofold coincidences while rotating the measurement settings locally.}
\label{Fig3}
\end{figure}
Interestingly, one can conclude from such a Bell test that the memory performs better that any possible classical strategies using for example a measure and prepare strategy or cloning followed by measurements in different basis. In this case, entanglement would be broken and the visibility would be limited to 1/3 assuming that the classical strategies introduce white noise on the singlet state.
Note that it has been shown recently that a device-independent certification is possible in the setup presented in fig.\ref{Fig3} \cite{sekatski_inprep}. In other words, it is possible to certify that the memory is a unitary operation and applies the identity on the qubits independently of the details and imperfections of the actual implementation by performing Bell tests with and without storage.
\\
We have derived and analyzed complementary criteria, both for continuous and discrete variables. They can be used as a benchmark to certify the quantum nature of the memory outcome. Our goal was to relate quantum optics measurements to experimentally accessible quantities that can be evaluated independently. This explicit criteria can be alternatively considered as a guide to anticipate the result of quantum measurements, to identify the limitations of an experimental setup and/or as an analytical tool for modeling a posteriori.
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