text:
\section{Introduction}

The purpose of this paper is to present an advanced use of the \emph{induced co- and contra-variant transform} illustrated by an example using the Heisenberg group \(\Space{H}{1}\). The induced covariant transform is produced by Gilmore--Perelomov coherent states~\citelist{\cite{Perelomov86} \cite{AliAntGaz14a}*{\S~7.1}}. It is well known that the transform is an intertwining operator to an induced representation, which explains our choice of  name for it. Although the topics of coherent states and covariant transform are extensively covered in existing  literature~\citelist{\cite{Kisil12b}  \cite{Grossmann86a}\cite{Grossmann85a}\cite{Perelomov86}\cite{AliAntGaz14a}\cite{Kirillov76}*{\S13}\cite{Kirillov04a}*{App.~V.2} \cite{Kisil17a} \cite{Berezin86}} we address some aspects which are commonly shadowed in the existing sources.

For example, the standard construction of induced representations~\citelist{\cite{Folland16a}*{ch.~6}
\cite{Kirillov04a}*{app.~v.2} \cite{KaniuthTaylor13a}} gives for the Heisenberg group the representation~\eqref{eq:H1-pol-rep-plane} below, which is different from~\eqref{FSB-Rep}---the celebrated Fock--Segal--Bargmann (FSB) representation~\cite{Folland89}*{\S~1.6} in the space of analytic function. Those two representations are linked by the \emph{peeling} map~(\S\ref{sec: Peeling}), which maps the annihilator of the representation space to the Cauchy--Riemann operator. The origin of the annihilator operator---the Lie derivative---in the properties of the fiducial vector is revealed in \S~\ref{Covariant Transform}. Section~\ref{Contravariant Transform} provides the respective consideration of the reconstructing vector for the contravariant  transform.

Overall, we present a machinery which allows to select fiducial and reconstructing vectors to ensure specific properties within the induced co- and contra-variant transforms. This technique does not rely on ad hoc knowledge and allows one to get  new insights even within the much-studied  framework of the Heisenberg group. As an elementary illustration, we apply this technique to find contra- and co-variant transforms, which  intertwine the coordinate and momentum representations of  \(\Space{H}{1}\), and predictably obtain the Fourier transform and its inverse. Less elementary  is the interpretation of the Zak transform as an induced covariant transform, which emerges as follows.

There are three forms of induction of representations of the Heisenberg group~\cite{Kirillov04a}*{\S~2.2}: the \emph{left quasi-regular} representation, the \emph{Schr\"odinger} representation and the \emph{lattice} representation (see \S~\ref{sec:induc-repr-heis}) for details.
A systematic use  of the \emph{covariant transform} to intertwine these three representations prompts the Zak transform and its inverse  as the covariant transform between the Schr\"odinger and lattice representations, with the \emph{Jacobi theta function} appearing as a vacuum state.
 Similarly, the \emph{pre-theta transform and its inverse} intertwine the (pre-)FSB and the lattice representations.

The classical Zak transform~\cite{Zak67a}, also known as~Weil--Brezin transform~\cite{Folland89}*{\S~1.10}, can be traced back to the works of Gelfand in 1950 (see also~\citelist{ \cite{Perelomov86}*{\S~1.5} \cite{Grochenig01a}*{ch.~8} \cite{Neretin11a}*{\S~8.1}}  for further applications and historical notes).
 This transform is an isometric isomorphism from \(\FSpace{L}{2}(\Space{R}{})\) onto \(\FSpace{L}{2}(\Space{T}{2})\) given, for \(f\in\FSpace{L}{2}(\Space{R}{})\), by
 \begin{equation}\label{Zak transform}
	[\oper{\tilde Z}f](u,v)=\sum_{n\in\Space{Z}{}} f(u+n)~\rme^{2\pi\rmi m n v}, \quad (u,v)\in\Space{T}{2}.
	\end{equation}
 Weil~\cite{Weil64a} defined the abstract Zak transform on arbitrary locally compact abelian (LCA) groups with respect to arbitrary closed subgroups.
Subsequently, the Zak transform was generalised several times, notably by Arefijamaal and Farashahi in~\cite{ArefijamaalGhaani13a} for semi-direct product groups \(G=K\rtimes H\), where \(H\) is a locally compact group and \(K\) is an LCA group.
 The connections between the Zak transform and (left) shift-invariant spaces were first observed by Joseph W. Iverson ~\cites{Iverson15a,Iverson18a} and, independently,  by D. Barbieri, E. Hern\'{a}ndez and V. Paternostro~\cite{BarbieriHernandezPaternostro15a}. Recently, the Zak transform has been developed for group representations by Joseph W. Iverson (~\cite{Iverson19a}). Its applications for time-frequency analysis can be found (~\cite{Grochenig01a}).
While the Zak transform has been widely generalised in  abelian and non-abelian setting \emph{for} the analysis of (left) shift invariant subspaces, its explicit interpretation as a covariant transform  appears to be new.

\section{Preliminaries on the Heisenberg group and its induced representations}
\label{sec:repr-homog-spac}

\subsection{The Heisenberg group and its Lie algebra}
\label{sec:heosenberg-group-its}
The \emph{polarised Heisenberg group} \(\Space[p]{H}{n}\) is the set of triples \((s,x,y)\), where \(s\in\Space{R}{}\) and \(x\), \(y\in \Space{R}{n}\), with the group law given as follows~\cite{Folland89}*{\S~1.2}:
\begin{equation}
 \label{eq:heisenberg-group}
 (s,x,y)\cdot(s',x',y')=(s+s'+xy',x+x',y+y').
\end{equation}
For the sake of simplicity in this paper, we will work with the one-dimensional case
 of \(\Space[p]{H}{1}\) and call it the Heisenberg group. It is a non-commutative group and its centre is a one-dimensional subgroup:
\begin{equation}
 \label{eq:heisenberg-centre}
 Z=\{(s,0,0)\in \Space[p]{H}{1}: \ s \in \Space{R}{}\} \, .
\end{equation}

The left action of \(\Space[p]{H}{1}\) on itself is given by
 \begin{equation}\label{left shift}
\tilde\Lambda(g):g'\mapsto g^{-1} g'.
\end{equation}
 We extend this action to a linear representation
\begin{equation}
 \label{eq:left-right-regular}
 \Lambda(g): F(g') \mapsto F(g^{-1}g'), \qquad g,g'\in\Space[p]{H}{1}
\end{equation}
on a certain linear space of functions on \(\Space[p]{H}{1}\)~\cite{Berndt07a}*{\S~1.1}.
The Lebesgue measure \(dg=ds\,dx\,dy\) on
 \(\Space[p]{H}{1}\sim\Space{R}{3}\) is a \emph{Haar measure} invariant under the left and right shifts. The action~\eqref{eq:left-right-regular} on the Hilbert space \(\FSpace{L}{2}(\Space[p]{H}{1},dg)\) of square-integrable functions on \(\Space[p]{H}{1}\)
is unitary~\cite{Kirillov04a}*{app.~v.2} and is called the \emph{left regular} representation.

Here, we introduce
\begin{equation}\label{Hiesen-lie-algebra-basis}
	S=(1,0,0),\quad X=(0,1,0),\quad Y=(0,0,1)
\end{equation}
which forms the basis of the Lie algebra \(\algebra{h}_1\) of \(\Space[p]{H}{1}\).
The commutator of \({X}\) and \({Y}\) is given by the celebrated \emph{Heisenberg commutation relation}:
\begin{equation}
  \label{eq:heisenberg-commutator}
	[{X},{Y}]= {S}.
\end{equation}
\index{representations of the Lie algebra}%

It is common for a representation \(\uir{}{}\) of a Lie group to pass to the derived representations \(\rmd \uir{}{}\) of the respective Lie algebra (cf.~\cite{Kirillov04a}*{ch.~2}).
Consider the derived representations of \(\algebra{h}_1\) spanned by:
  \begin{equation}
 \rmd\uir{S}{}=\frac{\rmd }{\rmd t} \uir{}{}(\rme^{t S})|_{t=0},\quad \rmd\uir{X}{}=\frac{\rmd }{\rmd t}\uir{}{}(\rme^{t X})|_{t=0},\quad \rmd\uir{Y}{}=\frac{\rmd }{\rmd t}\uir{}{}(\rme^{t Y})|_{t=0}.
	\end{equation}

If \(\uir{}{}\) is irreducible, \(\rmd\uir{S}{}\) is a multiple of the identity operator \(I\); that is, \(\rmd\uir{S}{}=-\rmi\myhbar I\) (cf.~\citelist{\cite{Kirillov04a}*{\S~2.2}  \cite{Bohm93}*{\S~II.3}}).

The important operators produced by the derived representations of \(\algebra{h}_1\) are \emph{ladder operators} (cf.~\citelist{\cite{Alamer19a}*{\S~5.3}\cite{Bohm93}*{\S~II.3}\cite{Kirillov04a}*{\S~2.2.1}}).
\begin{defn}
 \index{ladder operators ! creation and annihilation operators !ladder operator \(a^{\pm}\)
 \index{creation operator \(a^{+}\)}%
 \index{annihilation operator \(a^{-}\)
	 Let \(\kappa>0\) be some fixed number and \(\uir{}{}\) be a representation of \(\Space[p]{H}{}\) such that \(\uir{}{}(s,0,0)=\rme^{2\pi\rmi \myhbar s}I\). The \emph{ladder operators} are defined as follows:
	\begin{equation}\label{ladder-crea-anni}
          {a^-}{}=\frac{1}{\sqrt{2\myhbar\kappa}}(\kappa \rmi\,\rmd\uir{X}{}- \rmd\uir{Y}{}),\quad
          {a^+}{}=\frac{1}{\sqrt{2\myhbar\kappa}}(\kappa\rmi\,\rmd\uir{X}{} + \rmd\uir{Y}{}),
\quad \myhbar>0\,.
	\end{equation}
        The operators \(a^+\) and \(a^-\) are known as the {\em creation} and {\em annihilation} operators, respectively.
 \end{defn}
In this paper, we fix the parameter \(\kappa>0\) of the ladder operators \(a^{\pm}\)~\cite{AlmalkiKisil18a} and  indicate the dependence of ladder operators upon it. The Heisenberg commutator relation~\eqref{eq:heisenberg-commutator} implies
\begin{equation}\label{commutator}
  [a^-, a^+]=a^-a^+-a^+a^-=I.
\end{equation}
Also, for a unitary representation \(\uir{}{}\), we have \((a^{-1})^*=a^+\).

\begin{defn}\citelist{\cite{Alamer19a}*{\S~5.3}\cite{Kirillov04a}*{\S~2.6}} \label{de: vacuum}
 In the above notations, a vector \(\phi_0\in\FSpace{H}{}\) is called a {\em vacuum vector} if it is a null solution of the annihilation operator:
 \begin{displaymath}
 a^-\phi_0=0.
 \end{displaymath}
\end{defn}
We will need the following properties of the vacuum (see~\cite{Kirillov04a}*{\S~2.6}):
\begin{lem}
  \label{lem:vacuum-basic}
  If \(\phi_0\) is a vacuum of an irreducible representation \(\uir{}{}\), then:
  \begin{enumerate}
  \item \label{item:vacuum-uniqueness}
    \(\phi_0\) is unique up to a scalar multiple
  \item \label{item:vacuum-intertw}
    For an intertwining operator \(\oper{W}\) between \(\uir{}{}\) and another representation \(\uir{}{1}\), the image \(\oper{W}\phi_0\) is a vacuum for \(\uir{}{1}\).
  \end{enumerate}
\end{lem}

 \subsection{Induced representations of the Heisenberg group}
\label{sec:induc-repr-heis}

 We are interested in three  representations of \(\Space[p]{H}{1}\), which are \emph{induced by characters} of certain subgroups of \(\Space[p]{H}{1}\). Here, we briefly set the notations for this particular case of a  general theory of representations induced in the sense of Mackey (the reader is referred to~\cites{Kirillov76,Folland16a,KaniuthTaylor13a} for detailed presentations).

Let \(H\) be a subgroup of \(\Space[p]{H}{1}\) and \(\chi\) be a (complex unitary) character of \(H\). Let \(X=\Space[p]{H}{1}/H\) be the corresponding left \(\Space[p]{H}{1}\)-homogeneous space with the measure \(d x\), which factorises the Haar measure \(dg=dx\,dh\) for a Haar measure \(dh\) of \(H\). We write
\(\FSpace[\chi]{L}{2}(\Space[p]{H}{1})\) for the space of functions \(F(g)\) on
\(\Space[p]{H}{1}\) having these properties:
\begin{enumerate}
\item[(C)] \emph{\(H\)-covariance}:
 \begin{equation}
 \label{H-covariance property}
 F(gh)=\bar{\chi}_\myhbar (h)F(g), \qquad \text{ for all } g\in \Space[p]{H}{1}, \
 h\in H
\end{equation}
\item[(S)] \emph{\(\FSpace{L}{2}\)-summability} over \(X\):
\begin{equation}
 \label{eq:L2-condition1}
 \int_{X} \modulus{F(g)}^2dx<\infty,
\end{equation}
where the integral is meaningful since \(\modulus{F(g)}=\modulus{F(gh)}\) for all \(h\in H\) by \(H\)-covariance of \(F\).
\end{enumerate}
The space \(\FSpace[\chi]{L}{2}(\Space[p]{H}{1})\) is invariant under the left \(\Space[p]{H}{1}\)-shifts~\eqref{eq:left-right-regular} because the left and right shifts commute. An alternative realisation of the same representation is obtained from a given section \(\map{s}: \Space[p]{H}{1}/H \rightarrow \Space[p]{H}{1}\) which is a right inverse for the quotient map \(\map{p}: \Space[p]{H}{1} \rightarrow \Space[p]{H}{1}/H \); that is, \(\map{p}(\map{s}(w))=w\) for all \(w\in \Space[p]{H}{1}/H \). For such a section \(\map{s}\) and the character \(\chi\), we can define the lifting \(\oper{L}_\chi: \FSpace{L}{2}(\Space[p]{H}{1}/H) \rightarrow \FSpace[\chi]{L}{2}(\Space[p]{H}{1})\):
\begin{equation}
 \label{eq:lifting-definition}
 [\oper{L}_\chi f](g) = \bar{\chi}(r) f(\map{p}(g)),\qquad \text{ where } r = (\map{s}(\map{p}(g)))^{-1} g\,.
\end{equation}
Then, the lifting \(\oper{L}_\chi \) intertwines a representation \(\uir{}{\chi}\) on \(\FSpace{L}{2}(\Space[p]{H}{1}/H) \) and the left regular representation \(\Lambda\)~\eqref{eq:left-right-regular} restricted to \(\FSpace[\chi]{L}{2}(\Space[p]{H}{1})\):
\begin{displaymath}
\oper{L}_\chi \circ \uir{}{\chi}(g) = \left.\Lambda(g)\right|_{\FSpace[\chi]{L}{2}(\Space[p]{H}{1})}\circ \oper{L}_\chi, \qquad \text{ for all } g \in \Space[p]{H}{1}.
\end{displaymath}
Combining  the previous identities, we obtain an explicit expression for the induced representation:
\begin{equation}
  \label{eq:def-ind}
  [\uir{}{\chi}(g) f](x)= \bar{\chi}(\map{r}(g^{-1} * \map{s}(x)))\,
  f(g^{-1}\cdot x), \qquad \text{ where } f(x) \in \FSpace{L}{2}(\Space[p]{H}{1}/H).
\end{equation}

We now provide three forms of induced representations of \(\Space[p]{H}{1}\) acting on Hilbert spaces of square-integrable functions on homogeneous spaces, where the inductions are performed from characters of subgroups \(H\) of \(\Space[p]{H}{1}\) (cf~\cite{Alamer19a}*{\S~5.1}).

\begin{enumerate}
\item For the centre \(Z\)~\eqref{eq:heisenberg-centre} of \(\Space[p]{H}{1}\),
  the homogeneous space \(\Space[p]{H}{1}/Z\) is isomorphic to \( \Space{R}{2}\).
  Any real number \(\myhbar\neq 0\) defines
  the non-trivial character \(\chi_\myhbar(s,0,0)=e^{2\pi \rmi \myhbar s}\) of \(Z\).
  The respective maps are:
  \begin{equation}
    \label{eq:FSB-map-s-r}
    \map{p}(s,x,y)= (x,y), \qquad
    \map{s}(x,y)= (0,x,y), \qquad
    \map{r}(s,x,y)= (s,0,0)\,.
  \end{equation}

  The representation \(\Lambda_{\myhbar}\), which is induced from \(\chi_\myhbar\) on the space of square-integrable functions \(\FSpace{L}{2}(\Space{R}{2})\), is given by:
\begin{equation}
 \label{eq:H1-pol-rep-plane}
 	 [\Lambda_{\myhbar}(s,x,y)f] (x',y') = e^{2\pi \rmi \myhbar (s+x(y'-y))}f(x'-x,y'-y).
 \end{equation}
 This is called the \emph{left quasi-regular} representation. It is unitary reducible, and it can be decomposed into unitary irreducible components in many different ways (see~\cite{Alamer19a}*{\S 6.4}). For physical (aka mathematical)  reasons, the most popular irreducible module is the \emph{pre-Fock--Segal--Bargmann space} (\emph{pre-FSB space}):
\begin{equation}\label{pre-FSB}
	\FSpace{F}{\phi_{\myhbar\kappa}}(\Space{R}{2})=\{{f}:\quad (\kappa \rmi\,\linv{X}{}+ \linv{Y}{}) f = 0 \text{ and } f\in \FSpace{L}{2}(\Space{R}{}) \},
      \end{equation}
      where the \emph{Lie derivative} \(\linv{A}=\rmd R_{\myhbar}^{A}\) is the derivation of the  \emph{right quasi-regular} representation~\cite{Alamer19a}*{\S~5.4.2} on \(\FSpace{L}{2}(\Space{R}{2})\):
      \begin{equation}\label{right-action-center-rep}
        \begin{split}
          [R_\myhbar(g)f](x',y')=\rme^{-2\pi\rmi\myhbar(s+x'y)}f(x'+x,y'+y).
        \end{split}
      \end{equation}
      We will explain the origin of pre-FSB space in Subsection~\ref{Pre-Fock-Segal-Bargmann Transform}.

	\item For a two-dimensional maximal Abelian continuous subgroup of
 \(\Space[p]{H}{1}\):
	\begin{equation}
 \label{eq:H-prime-x-subgroup}
 H'_x=\{(s,0,y)\in\Space[p]{H}{1}: \ s,y\in\Space{R}{}\},
 \end{equation}
 the homogeneous space \(\Space[p]{H}{1}/ H'_x\) can be identified with \(\Space{R}{}\).
 The respective maps are:
 \begin{equation}
   \label{eq:maps-p-s-r-H_x-prime}
   \map{p}(s,x,y)=x,\ \qquad
   \map{s}(t) = (0,t,0),\qquad
   \map{r}(s,x,y) = (s-xy,0,y).
 \end{equation}
 The character \(\chi_\myhbar(s,0,y)=\rme^{2\pi \rmi \myhbar s}\) of \( H'_x\) induces the representation \(\uir{}{\myhbar}\) on a space of square-integrable functions, \(\FSpace{L}{2}(\Space{R}{})\),  is given by:
 \begin{equation}
 \label{eq:H1-schroedinger-rep}
 \textstyle[\uir{}{\myhbar}(s,x,y)f](t)= \rme^{2\pi \rmi \myhbar
 (s-ty)}\, f(t-x).
 \end{equation}
 This is the celebrated \emph{Schr\"odinger} representation, which is a unitary irreducible representation on \(\FSpace{L}{2}(\Space{R}{})\)~\cite{Folland89}*{ch.~1}.

 There is another two-dimensional maximal Abelian continuous subgroup of
 \(\Space[p]{H}{1}\):
 \begin{equation}
   \label{eq:H-prime-y-subgroup}
   H'_y=\{(s,x,0)\in\Space[p]{H}{1}: \ s,x\in\Space{R}{}\}.
 \end{equation}
 Subgroups \(H'_x\) and \(H'_y\) are conjugated by the automorphism of \(\map{i}: \Space[p]{H}{1} \rightarrow \Space[p]{H}{1}\):
 \begin{equation}
   \label{eq:Hp-automorphism}
   \map{i}(s,x,y)=(s-xy,-y,x)\,.
 \end{equation}
 There exist small but important differences between the respective maps cf.~\eqref{eq:maps-p-s-r-H_x-prime}:
 \begin{equation}
   \label{eq:maps-p-s-r-H_y-prime}
   \map{p}(s,x,y)=y,\ \qquad
   \map{s}(\lambda ) = (0,0,\lambda ),\qquad
   \map{r}(s,x,y) = (s,x,0).
 \end{equation}
 The character \(\chi_{\myhbar}(s,x,0)=\rme^{2\pi \rmi\myhbar s}\) of \(H_y'\) induced an alternative form of the Schr\"odinger representation:
\begin{equation}
 \label{eq:H1-schroedinger-rep-alt}
 \textstyle[\uir{\prime}{\myhbar}(s,x,y)f](\lambda )= \rme^{2\pi \rmi \myhbar
 (s+x(\lambda -y))}\, f(\lambda -y).
 \end{equation}
  In quantum mechanics,~\eqref{eq:H1-schroedinger-rep} and~\eqref{eq:H1-schroedinger-rep-alt} serve as \emph{coordinate} and \emph{momentum} representations of~\(\Space[p]{H}{1}\).
\item For a non-commutative discontinuous subgroup \(H_d\) of \(\Space[p]{H}{1}\):
	\begin{equation}\label{non-commuta}
H_d=\left\{(s, n, k):=(s ,n+\rmi k): (n, k)\in \Space{Z}{2}, s\in\Space{R}{}\right\}
\end{equation}
 we consider a homogeneous space \(\Space{T}{2}=\Space[p]{H}{1}/ H_d\).
 Let \(\chi_m(s,0,0)=e^{2\pi \rmi m s}\) be a character of \( H_d\).
 Let \(f(u,v)\) be a function of \(\FSpace{L}{2}(\Space{T}{2})\). We have two possibilities to treat \(f\):
 \begin{enumerate}
 \item \(f\) is a square-integrable and double quasi-periodic function on \(\Space{R}{2}\) (periodic in \(u\) and quasi-periodic in \(v\)), that is, for all \((n, k)\in \Space{Z}{2}\), we have
   \begin{equation}
     \label{eq: H_d-covariance property}
     f(u+n ,v+k)=\rme^{2\pi m\rmi u k}f(u,v);
   \end{equation}
\item \(f\) is a square-integrable function on the torus \(\Space{T}{2}=\lbrace (u,v): u, v\in[0,1) \rbrace\).
\end{enumerate}
If \(f\) is considered as a double quasi-periodic function on \(\Space{R}{2}\), the representation \(\uir{}{m}\) induced from the character \(\chi_m\) of \( H_d\) on \(\FSpace{ L}{2}(\Space{T}{2})\) is given by:
\begin{equation}
  \label{induced-smooth-rep-from-H_d}
  [\uir{}{m}(s,x,y)f](u,v)=\rme^{2\pi m\rmi(s+x(v-y))}f(u-x,v-y).
\end{equation}
The representation \(\uir{}{m}\) is unitary irreducible on \(\FSpace{ L}{2}(\Space{T}{2})\) and is called the \emph{lattice} representation~\cite{Cartier66a}.
The respective maps are:
\begin{equation}
  \label{eq:lattice-map-s-r}
  \map{p}(s,x,y)= (\{x\},\{y\}), \qquad
  \map{s}(u,v)= (s,u,v), \qquad
  \map{r}(s,x,y)= (s-\{x\}[y],[x],[y])\,,
\end{equation}
where \([x]\) is the integer part of \(x\) and \(\{x\}=x-[x]\) is the fractional part. Using them we express on \(\Space{T}{2}\) as follows:
\begin{equation}
  \label{eq:lattice-representation-T2}
  \begin{split}
    [\uir{m}{d}(s,x,y)f](u,v)&=e^{2\pi m\rmi (s+x\{v-y\}+u[v-y])} f(\{u-x\},\{v-y\})\\
    &=e^{2\pi m\rmi (s+x(v-y)+(u-x)[v-y])} f(\{u-x\},\{v-y\}) \\
  \end{split}
  \end{equation}
\end{enumerate}

For a unitary irreducible representation \(\uir{}{}\) of \(\Space[p]{H}{1}\), its restriction to the centre \(Z\) is a character \(\uir{}{}(s,0,0)= \rme^{2\pi \rmi \myhbar s}\), where the real parameter \(\myhbar\) is known as the Planck constant in a quantum mechanical context. The theorem of {Stone--von Neumann}~\cite{Folland89}*{\S~1.5} states that any two infinite-dimensional
irreducible strongly continuous unitary representations of \(\Space[p]{H}{1}\) with the same Planck constant are unitary equivalent.
 In this paper, motivated by the physical framework and for the sake of simplicity, we only consider the positive Planck constant \(\myhbar>0\).

\section{The Covariant transform on \(\Space[p]{H}{1}\)}
\label{Covariant Transform}

\subsection{An Induced covariant transform}
\label{sec:induc-covar-transf}

In this work we need an extended version of the covariant transform which covers the Banach space situation~\citelist{\cite{Kisil98a}*{\S~2} \cite{Kisil13a}}. All representations in this paper are assumed to be strongly continuous.
\begin{defn}
	Let \(\uir{}{}\) be a representation of a group in a space \(V\), and \(F\) be an operator from \(V\) to a space \(U\). We define a \emph{covariant transform} \(\oper{W}_F\) from \(V\) to the space \(L(G,U)\) of a \(U\)-valued function on \(G\) by the formula:
	\begin{equation}\label{gen-cov}
		\oper{W}_F:v\mapsto\tilde v=F(\uir{}{}(g^{-1})v),\quad v\in V,\, g\in G.
	\end{equation}
\end{defn}
The fundamental property of the covariant transform \(\oper{W}_F\)~\eqref{gen-cov} is that \(\oper{W}_F\) intertwines the representation \(\uir{}{}\) and the \emph{left regular action} \(\Lambda\) of \(G\):
\begin{equation}
 \label{eq:left-shift-itertwine}
 \oper{W}_F\circ \uir{}{}(g) =\Lambda(g)\circ \oper{W}_F \quad\text{ for all } g\in G,
\end{equation}
where \(\Lambda(g): f(g') \mapsto f(g^{-1}g')\), \(g,g'\in G\).

For the Gilmore--Perelomov coherent states~\citelist{ \cite{Perelomov86} \cite{AliAntGaz14a}*{\S~7.1.2}}, it is enough to have the covariant transform values on a homogeneous space rather than the entire group. We name it \emph{the induced covariant transform}~\cite{Kisil13a} due to its connections with induced representations (cf.~\eqref{intertwinging-irre-repr}). More specifically, for the Heisenber group it is defined as follows.
Let \(\uir{}{}\) be an irreducible unitary representation of \(\Space[p]{H}{1}\) on a Hilbert space \(\FSpace{H}{}\), and \(H\) be a closed subgroup of \(\Space[p]{H}{1}\). Let \(X=\Space[p]{H}{1}/H\) be a homogeneous space. Let \(\phi_0\in \FSpace{H}{}\) be a \emph{fiducial vector}, that is:
\begin{equation}
 \label{joint eigenvector}
		\uir{}{}(h)\phi_0=\chi(h) \phi_0, \quad\text{for all}~ h\in H,
	\end{equation}
for some character \(\chi\) of \(H\).
 The induced covariant transform \(\oper{W}_{\phi_0}^{\uir{}{}}\) is a map from the Hilbert space \(\FSpace{H}{}\) to a space \(W(X)\) of functions on \(X=\Space[p]{H}{1}/H\) given as follows:
 \begin{equation}
 \label{eq:induce-wavelet-transform-1}
 	\oper{W}_{\phi_0}^{\uir{}{}}: f \mapsto \tilde{f}(\map{s}(x)) = \scalar{f}{\uir{}{}(\map{s}(x))\phi_0},\qquad
 \quad x\in X,
\end{equation}
where \(\map{s}:X\mapsto \Space[p]{H}{1}\) is a
Borel section (the right inverse of the natural projection \(\map{p}:\Space[p]{H}{1}\rightarrow
\Space[p]{H}{1}/H\)).

The main algebraic property of the induced covariant transform~\eqref{eq:induce-wavelet-transform-1} is that it
intertwines \(\uir{}{}\) on \(\FSpace{H}{}\) with a representation \(\uir{}{\chi}\) on \(W(X)\) induced by the character \(\chi\) of the subgroup \(H\). That is,
\begin{equation}\label{intertwinging-irre-repr}
	\uir{}{\chi}\circ \oper{W}_{\phi_0}^{\uir{}{}}=\oper{W}_{\phi_0}^{\uir{}{}}\circ \uir{}{}.
	\end{equation}
Alternatively, this can be observed by the fact that any function of the image of the induced covariant transform~\eqref{eq:induce-wavelet-transform-1} has the \(H\)-covariance property
\begin{equation}\label{eq:prove-H-covarianc}
	\tilde f(gh)=\bar \chi(h)\tilde f(g).
\end{equation}
The main analytic property of the induced covariant transform is formulated in terms of \emph{the matrix coefficient}:
 \begin{equation} \label{matrix coefficient}
 	\oper{W}(f,\phi)(x,y)=\scalar{f}{\rho (0,x,y)\phi}\, \qquad \text{ for } f, \phi\in \FSpace{H}{}
 \end{equation}
which is a continuous linear map \(	\FSpace{H}{} \times\FSpace{H}{} \rightarrow \FSpace{L}{2}(\Space{R}{2})\).
Moreover,  the map~\eqref{matrix coefficient} is \emph{sesqui-unitary}, that is  for all \(f_1,\phi_1,f_2,\phi_2\in\FSpace{H}{}\),
 \begin{equation}\label{generaL-sesqui-unitary}
   \scalar{\oper{W}(f_1,\phi_1)}{\oper{W}(f_2,\phi_2)}_{\FSpace{L}{2}(\Space{R}{2}) }=\scalar{f_1}{f_2}_{\FSpace{H}{}}\overline{\scalar{\phi_1}{\phi_2}}_{\FSpace{H}{}}.
 \end{equation}

 \begin{example}[The inverse Fourier transform]
   \label{ex:covariant-Fourier}
   To find the covariant transform which intertwines two forms of the Schr\"odinger representations~\eqref{eq:H1-schroedinger-rep} and~\eqref{eq:H1-schroedinger-rep-alt}, we shall take the fiducial vector which would be the eigenfunction for all representations
\(\uir{}{\myhbar}(s,x,0)\)~\eqref{eq:H1-schroedinger-rep}. That is
   \begin{displaymath}
     [\uir{}{\myhbar}(s,x,0)f](t)
     = \rme^{2\pi \rmi \myhbar s}\, f(t-x)
     = \rme^{2\pi \rmi \myhbar (s)} f(t)\,.
   \end{displaymath} Of course the only solution is the constant function \(f(t)\equiv c\). Then the corresponding induced covariant transform based on the maps~\eqref{eq:maps-p-s-r-H_y-prime} is:
 \begin{equation}
   \label{eq:Fourier-transform}
   \begin{split}
     [\oper{W} f](y) & = c \int_{\Space{R}{}} f(t)\, \rme^{2\pi \rmi \myhbar y t} \,\rmd t\,.
   \end{split}
 \end{equation}
 Of course, this transformation is the (inverse) Fourier transform which intertwines the representations \eqref{eq:H1-schroedinger-rep} and~\eqref{eq:H1-schroedinger-rep-alt}. The significance of this intertwining property for the harmonic analysis is revealed in the work of Howe ~\cite{Howe80a}.
 \end{example}

Although many functions can be taken as fiducial vectors, some of them turn out to be much more preferable. The origin of their advantages is revealed by the following observation~\cite{Kisil11c}*{\S~5}. Let \(G\) be a Lie group and \(\uir{}{}\) be its representation in a Hilbert
space \(\FSpace{H}{}\). Let \([\oper{W}_\phi f](g)=\scalar{f}{\uir{}{}(g)\phi}\) be the
covariant transform defined by a fiducial vector \(\phi\in \FSpace{H}{}\). Then, the covariant transform
intertwines \emph{right shifts} \(R(g): f(g') \mapsto f(g'g)\) on the group \(G\) with the
associated action \(\uir{}{}\) on fiducial vectors:\begin{equation}
 \label{eq:wave-intertwines-right}
 R(g) \circ \oper{W}_\phi= \oper{W}_{\uir{}{}(g)\phi}.
\end{equation}
There are many interesting applications of this simple observation~\cites{Kisil11c,Kisil12d,Kisil17a,AlmalkiKisil18a,AlmalkiKisil19a}, in particular:
\begin{prop}
  \label{co:cauchy-riemann-integ} \cites{Kisil11c,Kisil12d}
Let \(G\) be a Lie group with a Lie algebra \(\algebra{g}\) and \(\uir{}{}\) be a representation of \(G\) on a Hilbert space \(\FSpace{L}{2}(\Space{R}{n})\). We denote the derived representation of \(\uir{}{}\) by \(\rmd\uir{X}{}\). Let \(\phi\) be a fiducial vector in the Schwartz space \(\oper{S}(\Space{R}{n})\) such that
 \((\sum_{j=1}^n a_j\rmd\uir{X_j}{})\phi=0\), for some \(a_j\in\Space{C}{}\). Then, the image of the covariant transform consists of  functions \(f\) such that:
 \begin{equation
 \label{eq:dirac-op}
 \left(\sum_{j=1}^n\bar{a}_j \linv{X_j}\right)\,\tilde f =0,
 \end{equation}
 where \(\linv{X}\) denotes the \emph{Lie derivative}---the derivation  of the right regular representation \(R\) of \(G\):
 \begin{equation}
   \label{eq:Lie-derivarive}
   \linv{X} = \left. \frac{\rmd R(\rme^{tX})}{\rmd t} \right|_{t=0}\,, \qquad \text { for } X\in\algebra{g}.
 \end{equation}
\end{prop}

An induced covariant transform from a representation \(\uir{}{}\) on a Banach space \(\FSpace{B}{}\) can be defined in a similar fashion. Let \(H\) be a closed subgroup of \(\Space[p]{H}{1}\) and \(X=\Space[p]{H}{1}/H\) be a homogeneous space.	 Consider a continuous section \(\map{s}:\Space[p]{H}{1}/H\rightarrow \Space[p]{H}{1}\), which is the right inverse of the natural projection \(\map{p}:\Space[p]{H}{1}\rightarrow
\Space[p]{H}{1}/H\). {Denote by} \(B^*\) {the dual space of the Banach space} \(B\) and \(\uir{}{}^*\) the {adjoint operator to} \(\uir{}{}\). Let \(l_0\in B^*\) {be a non-zero test function such that}
 \begin{equation}\label{joint eigenvector-test}
		\uir{}{}^*(h)l_0=\bar\chi(h) l_0, \quad\text{for all}~ h\in H,
	\end{equation}
for some character \(\chi\) of \(H\).
{The induced covariant transform} \(\oper{W}_{l_0}\)  is defined as~\citelist{\cite{Kisil13a}\cite{Kisil98a}*{\S~2}}:
 	\begin{equation}
 	\label{wavelet_B}
 	\tilde { v}(x) =[\oper{W}_{l_0}v](x)= \scalar{\uir{}{}(\map{s}(x)^{-1})v}{l_0}=\scalar{v}{\uir{}{}^*(\map{s}(x))l_0}.
 	\end{equation}
Similarly to the Hilbert space case~\eqref{intertwinging-irre-repr}, the induced covariant transform~\eqref{wavelet_B}
intertwines \(\uir{}{}\) on \(B\) {with a representation} \(\uir{}{\chi}\) on \(W(X)\) {induced by the character} \(\chi\) of the subgroup \(H\)~\cite{Kisil98a}*{\S~2, pro. 2.6}.

Thus the induced covariant transform can be used as
\begin{itemize}
\item Reproducing formulae on representation spaces; and
\item Intertwining operators between different realisations of equivalent representations.
\end{itemize}
Both applications will be illustrated below.

\subsection{The (pre-)Fock--Segal--Bargmann transform}\label{Pre-Fock-Segal-Bargmann Transform}
We look for an induced covariant transform \( \oper{W}_\phi^{\uir{}{\myhbar}}: \FSpace{L}{2}(\Space{R}{}) \rightarrow \FSpace{L}{2}(\Space{R}{2})\), which intertwines the Schr\"odinger representation~\eqref{eq:H1-schroedinger-rep} and the left quasi-regular representation restricted to an irreducible component of \(\FSpace{L}{2}(\Space{R}{2})\).
 In fact,
for the character
\(\chi_\myhbar(s,0,0)= \rme^{2\pi\rmi\myhbar s}\) of the centre \(Z=\{(s,0,0)\in\Space[p]{H}{1}:\
s\in\Space{R}{}\}\), any vector
\(\phi\in\FSpace{L}{2}(\Space{R}{})\) satisfies the following specialisation of~\eqref{joint eigenvector}
\begin{equation*}
	\uir{}{\myhbar}(s,0,0)
\phi=\chi_\myhbar(s,0,0) \phi,\quad \text{for all}~ (s,0,0)\in Z.
\end{equation*}
Let \(\map{s}: \Space[p]{H}{1}/Z\rightarrow \Space[p]{H}{1}: (x,y)\mapsto (0,x,y)\) be a continuous section~\cite{Alamer19a}*{\S 5.1.2}.
Thus, for all \(f\in\FSpace{L}{2}(\Space{R}{})\), the induced covariant transform \(\oper{W}_{\phi}^{\uir{}{\myhbar}}\) for any fiducial vector \(\phi\in\FSpace{L}{2}(\Space{R}{})\) is:
 \begin{equation}
 \label{eq:left-transform}
 [\oper{W}_{\phi}^{\uir{}{\myhbar}}(f)](x,y)=\int_{\Space{R}{}} f(t)\, \rme^{2\pi\rmi\myhbar ty}\, \phi(t-x)\,\rmd t.
 \end{equation}
The main properties of \(\oper{W}_\phi^{\uir{}{\myhbar}}\)
 follow from the general properties of the covariant transform.
\begin{cor} \label{properties-pre-transform}
  Let \(\phi\in\FSpace{L}{2}(\Space{R}{})\) be a fiducial vector such that \(\norm{\phi}=1\).
  The covariant transform \(\oper{W}_\phi^{\uir{}{\myhbar}}: \FSpace{L}{2}(\Space{R}{}) \rightarrow \FSpace{L}{2}(\Space{R}{2})\) is
  a unitary intertwining operator between the Schr\"odinger representation \(\uir{}{\myhbar}\) on \(\FSpace{L}{2}(\Space{R}{})\) and the left quasi-regular representation \(\Lambda_{\myhbar}\) restricted on the image space
  \begin{equation}\label{image-covariant-functions-space}
    F_\phi(\Space{R}{2}):=\{\oper{W}_{\phi}^{\uir{}{\myhbar }}(f): f\in \FSpace{L}{2}(\Space{R}{})\}.
  \end{equation}
  In particular, \(\Lambda_{\myhbar}\) is an irreducible representation on \(F_\phi(\Space{R}{2})\).
\end{cor}

So far all fiducial vectors seem to be equally suitable, yet its is common to give the strong preference to a vacuum vector of the Schr\"odinger representation---the Gaussian:
\begin{equation}
  \label{eq:Gaussina-defn}
  \phi_{\myhbar\kappa}(t)=2^{1/4}\rme^{-\frac{\pi\myhbar}{\kappa} t^2}\,.
\end{equation}
This preference is explained by Prop.~\ref{co:cauchy-riemann-integ}: since  the Gaussian is a null-solution
 to the annihilation operator
 \begin{equation}\label{annihilation-schrodinger}
 	a^-_{\uir{}{\myhbar}}=\rmd\uir{\kappa X-\rmi Y}{\myhbar}=-2\pi\myhbar t-\kappa\partial_t\,,
 \end{equation}
 the image space of the induced covariant transform \(\oper{W}_{\phi_{\myhbar\kappa}}^{\uir{}{\myhbar}}\)~\eqref{eq:induce-wavelet-transform-1}  is annihilated by the Lie derivative  \(\linv{\kappa X+\rmi Y}\) (the derived representation from the right regular action \(R_\myhbar\)). This allows to give an intrinsic characterisation of the image space.

 Explicitly, \(\oper{W}_{\phi_{\myhbar\kappa}}^{\uir{}{\myhbar}}\) is given by:
 \begin{equation}
 \label{eq:prefsb-transform}
 \begin{split}
\tilde{f}(x,y)&:=[\oper{W}_{\phi_{\myhbar\kappa}}^{\uir{}{\myhbar}}f](x,y)=\scalar{f}{\uir{}{\myhbar}(0,x,y)\phi_{\myhbar\kappa}}
 \\&=\left(\frac{\myhbar}{\kappa}\right)^{1/2}2^{1/4}\int_{\Space{R}{}} f(t)\, \rme^{2\pi\rmi\myhbar ty}\, \rme^{-\frac{\pi \myhbar}{\kappa} (t-x)^2}\,\rmd t\,,
 \end{split}
\end{equation}
where  the measure is renormalised by the factor \(({\frac{\myhbar}{\kappa}})^{1/2}\).
For reasons explained here, we call it the \emph{pre-FSB transform}~\citelist{\cite{Neretin11a}*{\S~4.2}\cite{Folland89}*{\S~1.6}} from \(\FSpace{L}{2}(\Space{R}{})\) into the \emph{pre-FSB space} \(\FSpace{F}{\phi_{\myhbar\kappa}}(\Space{R}{2})\)~\eqref{pre-FSB}.
The image space \(\FSpace{F}{\phi_{\myhbar\kappa}}(\Space{R}{2})\) is a subspace of square-integrable functions on \(\Space{R}{2}\). The left quasi-regular representation \(\Lambda_{\myhbar}\) restricted on the pre-FSB space \(\FSpace{F}{\phi_{\myhbar\kappa}}(\Space{R}{2})\) is called the \emph{pre-FSB representation}.
The prefix ``pre-'' is removed by a unitary
operator---the \emph{peeling}, which will produce the \emph{FSB space} of analytic functions on \(\Space{C}{}\) in Subsection~\ref{exm:peeling-rep-center-FSB}.

 \subsection{The Zak transform}\label{exp:calculating-Zak}
 In this subsection, we derive the Zak transform~\citelist{\cite{Folland89}*{ch~4}\cite{Cartier66a}\cite{Berndt07a}*{ch~9}} as a particular case of the covariant transform. More specifically, we look for  the induced covariant transform
 \begin{equation}
   \oper{W}_{l_0}^{\uir{}{\myhbar }}:\FSpace{L}{2}(\Space{R}{})\rightarrow \FSpace{L}{2}(\Space{T}{2}),
 \end{equation}
 which intertwines the Schr\"odinger representation~\eqref{eq:H1-schroedinger-rep} and the lattice representation~\eqref{induced-smooth-rep-from-H_d}.
For an integer \(m\), let \(\chi_m(s,n.k)=\rme^{2 \pi \rmi m s}\) be the character of the subgroup \(H_d=\{(s,n,k):~s\in \Space{R}{}, n,k\in\Space{Z}{}\}\)~\eqref{non-commuta} of \(\Space[p]{H}{1}\).
A required fiducial vector \(l_0\) shall satisfy~\eqref{joint eigenvector}, which  in this setup becomes

\begin{equation}\label{satifying-con-zak}
  \begin{split}
    \rme^{2\pi \rmi \myhbar s} \rme^{-2\pi \rmi \myhbar t k}~l_0(t-n)=\rme^{2\pi \rmi m s} l_0(t)\,, \qquad \text{ for all } (s,n,k)\in H_d.
  \end{split}
\end{equation}
 The left- and the right-hand sides of~\eqref{satifying-con-zak} are equal if and only if
 \begin{enumerate}
 \item \(\myhbar=m\) [from considering \((1,0,0)\in  H_d\)];
 \item the function \(l_0\) is a periodic function (with the period \(1\))  [from considering \((0,1,0)\in  H_d\)]; and
 \item \(\mathrm{supp}(l_0)\subseteq\Space{Z}{}\)   [from considering all \((0,0,k)\in  H_d\)].
 \end{enumerate}
 Of course, the last two conditions imply  that \(\mathrm{supp}(l_0)=\Space{Z}{}\).
The simplest non-zero vector \(l_0\) satisfying~\eqref{satifying-con-zak} would be the
\emph{Dirac comb} distribution, that is,
\begin{equation}
  \label{eq:Dirac-comb}
  l_0(t)=\sum_{n\in\Space{Z}{}}\delta(t-n)\,,
\end{equation}
which  is a periodic distribution constructed from the \emph{Dirac delta} \(\delta(t)\).

\begin{rem}
{Let} \(K\) {be a compact subset of} \(\Space{R}{}\), and \(\FSpace{C}{}(K)\) {be the space of continuous functions on} \(\Space{R}{}\) {supported in} \(K\), {which is a Banach space equipped with  the uniform norm. Let} \(\FSpace{C}{c}(\Space{R}{})\) {be the union of these Banach spaces, where} \(\FSpace{C}{c}(\Space{R}{})\) {inherits a natural inductive limit topology} (\citelist{\cite{Rudin91a}\cite{Folland16a}}). We denote by \(\FSpace[*]{C}{c}(\Space{R}{})\) {the space of all continuous functionals (\emph{pseudomeasures}) on} \(\FSpace{C}{c}(\Space{R}{})\), {which is
 the intersection of all duals of} \(\FSpace{C}{}(K)\). {As the Dirac comb} \(l_0(t)\)~\eqref{eq:Dirac-comb} {is a finite measure in any compact set \(K\), thus} \(l_0\in \FSpace[*]{C}{c}(\Space{R}{})\) {is a pseudomeasure. Therefore, we
 consider the Schr\"odinger representation} \(\uir{}{\myhbar}\) of \(\Space[p]{H}{1}\) on \(\FSpace{C}{c}(\Space{R}{})\subset \FSpace{L}{2}(\Space{R}{})\)
{to be restricted to the Banach space} \(\FSpace{C}{}(K)\), {for any compact subset} \(K\) of \(\Space{R}{}\).
\end{rem}
 Let \(\map{s}: \Space{T}{2}\rightarrow \Space[p]{H}{1}:\, (u,v)\mapsto (0,u,v)\) be a continuous section (cf~\cite{Alamer19a}*{\S~5.1.3}). Let \(\FSpace{C}{c}(K)\) be the space of smooth functions that are compact support in \(K\). For \(f\in \FSpace{C}{c}(K)\subset\FSpace{L}{2}(\Space{R}{})\), we calculate the induced covariant transform as follows:
 \begin{align}
\nonumber
     [\oper{W}_{l_0}^{\uir{}{\myhbar }}(f)](u,v)
     &=\scalar{f}{\uir{}{\myhbar}(0,u,v )l_0}
\\
\nonumber
&=\int_{\Space{R}{}} f(t)\, \rme^{2\pi\rmi \myhbar t v }\, \bar l_0(t-u)\,\rmd t
\\
\nonumber
&=\int_{\Space{R}{}} f(t)\, \rme^{2\pi \rmi m t v}\sum_{n\in\Space{Z}{}}\delta(t-(u+n))\,\rmd t,\quad \myhbar=m
\\
\nonumber
&=\sum_{n\in\Space{Z}{}}\int_{\Space{R}{}} f(t)\, \rme^{2\pi \rmi m t v }\delta(t-(u+n))\,\rmd t
\\
\nonumber
&=\sum_{n\in\Space{Z}{}} f(u+n)\, \rme^{2\pi \rmi m v (u+n)}
   \\
\label{eq:covariant-Zak1}
     &=\rme^{2\pi \rmi m u v}\sum_{n\in\Space{Z}{}} f(u+n)\, \rme^{2\pi \rmi m v n}.
 \end{align}
 This is the Zak transform~\citelist{\cite{Folland89}*{Ch~4}\cite{Cartier66a}\cite{Berndt07a}*{Ch~9}} up to the factor \(\rme^{2\pi \rmi m u v}\). Following previous research~\cite{Alamer19a}, we call~\eqref{eq:covariant-Zak1} the \emph{co-Zak transform}.
\begin{rem}\label{extended_Zak}
	{Since the co-Zak transform is defined on} \(\FSpace{C}{c}(\Space{R}{})\), {which is dense on} \(\FSpace{L}{2}(\Space{R}{})\), {the co-Zak transform can be extended to be defined on the entire} \(\FSpace{L}{2}(\Space{R}{})\).
\end{rem}
 The general properties of the covariant transform \(\oper{W}_{l_0}^{\uir{}{\myhbar }}\) yield corresponding properties of the Zak transform.
 \begin{cor} \label{ZAk-proper}
   For \(f\in\FSpace{L}{2}(\Space{R}{})\), let
   \begin{equation*}
      [\oper{Z}f](u,v) \colonequals
  [\oper{W}_{l_0}^{\uir{}{\myhbar }}(f)](u,v)=\rme^{2\pi \rmi m u v}\sum_{n\in\Space{Z}{}} f(u+n)~\rme^{2\pi\rmi m n v}
\end{equation*}
be the co-Zak transform~\eqref{eq:covariant-Zak1}.
 Then, we have the following properties:
 \begin{enumerate}
 \item \label{cor:interwt-zak} The operator \(\oper{Z}:\FSpace{L}{2}(\Space{R}{})\rightarrow \FSpace{L}{2}(\Space{T}{2})\) intertwines the Schr\"odinger representation \(\uir{}{\myhbar}\) and the lattice representation \(\uir{}{m}\). That is, \(\textstyle\uir{}{m}\circ \oper{Z}=\oper{Z}\circ\textstyle\uir{}{\myhbar}\), for \(\myhbar=m\).

 \item \label{unit-zak} The operator \(\oper{Z}:\FSpace{L}{2}(\Space{R}{})\rightarrow \FSpace{L}{2}(\Space{T}{2})\) is unitary.

 \item \label{cor:period-zak} The image space of \([\oper{Z}f](u,v)\) consists of functions \(\tilde f(u,v)\) that have the double-quasi-periodic property on \(\Space{R}{2}\).
 \end{enumerate}
\end{cor}
\begin{proof}

\begin{enumerate}
\item Since \(\oper{W}_{l_0}^{\uir{}{\myhbar }}=\oper{Z}\), the intertwining property in~\eqref{eq:left-shift-itertwine}, for \(\myhbar=m\), implies
  \begin{equation}
    \uir{}{m}\circ \oper{W}_{l_0}^{\uir{}{\myhbar}}=\oper{W}_{l_0}^{\uir{}{\myhbar}}\circ \uir{}{\myhbar}.
  \end{equation}

\item Since \(\oper{W}_{l_0}^{\uir{}{\myhbar }}\) is an intertwining operator between two irreducible representations of \(\Space[p]{H}{1}\),  by \emph{Schur's Lemma}~\cite{Kirillov76}*{Thm.~8.2.1}, the covariant transform \(\oper{W}_{l_0}^{\uir{}{\myhbar }}\) is a bijection \(\FSpace{L}{2}(\Space{R}{})\rightarrow \FSpace{L}{2}(\Space{T}{2})\).
 ``When you have an intertwining operator between irreducible
representations, it has to be terrible not to be unitary''~\cite{Howe80a}, if a right factor is used. The scaling can be checked from the identity
\begin{displaymath}
  \norm[\FSpace{L}{2}(\Space{R}{})]{\chi_{[0,1]}} = \norm[\FSpace{L}{2}(\Space{T}{2})]{\oper{W}_{l_0}^{\uir{}{\myhbar }}\chi_{[0,1]}}
\end{displaymath}
for the indicator function \(\chi_{[0,1]}\) of the interval \([0,1]\) with \(\oper{W}_{l_0}^{\uir{}{\myhbar }}\chi_{[0,1]}(u,v)=\rme^{2\pi \rmi m (u+1)v}\).
\item The image space of the induced covariant transform \(\tilde f= \oper{W}_{l_0}^{\uir{}{\myhbar }}(f)\) has the \(H\)-covariance property~\eqref{eq:prove-H-covarianc} \(
		\tilde{f}(gh)= {\bar\chi}(h)\tilde{f}(g)\),
for all \( g \in G\) and \( h\in H\).
For the subgroup \(H_d=\left\{(s, n, k)=(s ,n+\rmi k): s\in\Space{R}{},~ n, k\in \Space{Z}{}\right\}\), it has exactly the double-quasi-periodic property~\eqref{eq: H_d-covariance property}:
 \begin{equation}
 \tilde{f}(u+n ,v+k)=\rme^{2\pi m\rmi u k}\tilde{f}(u,v).
 \end{equation}
\end{enumerate}

\end{proof}

\begin{example}[the pre-theta function]
  It is natural to evaluate the covariant transform of the vacuum vector---Gaussian \(\phi_{\myhbar\kappa}\)~\eqref{eq:Gaussina-defn} with \(\myhbar=m\):

  \begin{equation}
    \label{calculating-theta-vacuum}
 	\begin{split}
          \Phi_{m\kappa}(u,v)
          &:= [{\oper{Z}}\phi_{ m \kappa}](u,v
          \\
          &= \rme^{\pi \kappa  (3\omega^2-\bar{\omega}^2-2\omega \bar{\omega}) / (4m)}\, \sum_{n\in\Space{Z}{}}\rme^{-\frac{\pi m}{\kappa}n^2}\,\rme^{ 2\pi\rmi n  \omega}\\
          &=\rme^{\pi \kappa  (3\omega^2-\bar{\omega}^2-2\omega\bar{\omega})/(4m)}\,\Theta_{m\kappa}\left( \omega, \frac{\rmi m }{\kappa }\right)
          ,
	 	\end{split}
 \end{equation}
 where \(\omega=m(v+\rmi u/\kappa )\in\Space{C}{}\)  and \(\Theta_{m\kappa}\) is the \emph{Jacobi theta function}. We will use the notation \(\Phi_{m\kappa}(\omega ,\bar{\omega})=\Phi_{m\kappa}(u,v)\) as well.
 By  Lem.~\ref{lem:vacuum-basic}(\ref{item:vacuum-intertw}) \(\Phi_{m\kappa}\), is a vacuum of the lattice representation.
\end{example}

\subsection{The (pre-)theta transform}\label{integration-transform-theta}
In the present subsection, we look for an intertwining operator
\begin{equation}
	\oper{W}_{\Phi}^{\uir{}{m}}:\FSpace{L}{2}(\Space{T}{2})\rightarrow \FSpace{L}{2}(\Space{R}{2})
\end{equation}
 between the lattice representation and the left quasi-regular representation restricted to an irreducible component of \(\FSpace{L}{2}(\Space{R}{2})\).
Although the formula of the left quasi-regular representation~\eqref{eq:H1-pol-rep-plane}
looks very similar to the lattice representation's formula~\eqref{induced-smooth-rep-from-H_d}:
 they act on different spaces \(\FSpace{L}{2}(\Space{R}{2})\) and \(\FSpace{L}{2}(\Space{T}{2})\), respectively.

 Let \(\chi_\myhbar(s,0,0)= \rme^{2\pi\rmi\myhbar s}\) be the character of the centre \(Z\subset \Space[p]{H}{1}\).
As was already mentioned (Subsection~\ref{Pre-Fock-Segal-Bargmann Transform}), any vector
\(\Phi\in\FSpace{L}{2}(\Space{T}{2})\) satisfies version \(\uir{}{m}(s,0,0)
\Phi=\chi_\myhbar(s,0,0) \Phi\) of~\eqref{joint eigenvector}, for all \((s,0,0)\in Z\) and \(\myhbar=m\). Thus, the respective covariant transform \(\oper{W}_{\Phi}\) intertwines the lattice and quasi-regular representations.

We may be more specific and request that \(\oper{W}_{\Phi}\) map \(:\FSpace{L}{2}(\Space{T}{2})\) to the pre-FSB space \(\FSpace{F}{2}(\Space{R}{2})\).
As in the case of the pre-FSB transform, the vacuum \(\Phi_{m\kappa}\)~\eqref{calculating-theta-vacuum} shall be taken as the fiducial vector. Indeed, by Prop.~\ref{co:cauchy-riemann-integ}, the image space of \(\oper{W}_{\Theta}^{\uir{}{m}}\) is annihilated by the right ladder \(\rmi \kappa\linv{ X}+\linv{Y}\)~\eqref{right-action-center-rep}. Thus,
the induced covariant transform \(\oper{W}_{\Phi}^{\uir{}{m}}:\FSpace{L}{2}(\Space{T}{2})\rightarrow \FSpace{F}{2}(\Space{R}{2})\) is calculated (up to normalisation) as follows:
\begin{equation
   \label{theta-transform-1}
  \begin{split}
    [\oper{W}_{\Phi}^{\uir{}{m}}(f)](x,y)
    &=\scalar{f}{\uir{}{m}(\map{s}(x,y) )\Phi_{m\kappa}
    \\
    &=\int_{\Space{T}{2}} f(u,v)\, \rme^{-2\pi\rmi m x(v-y)}\, \bar{\Phi}_{m\kappa}(u-x,v-y)\,\rmd u\,\rmd v\\
        &=\int_{\Space{T}{2}} f(\omega )\, \rme^{\pi \kappa[2(\zeta-\bar{\zeta})[(\bar{\omega}-\bar{\zeta})+(\omega-\zeta)]+3 (\bar{\omega}-\bar{\zeta})^2-(\omega-\zeta)^2-2(\omega-\zeta)(\bar{\omega}-\bar{\zeta})]/(4 m)
}\\
    &\qquad\qquad \qquad\times
    \overline{\Theta_{m\kappa}\left(\omega -\zeta,\frac{\rmi m}{\kappa }\right)}\,\frac{\kappa\, \rmd \omega \wedge\rmd \bar{\omega}}{\rmi m }
 \end{split}
\end{equation}
where \(\omega=m(v+\rmi u/\kappa )\) and \(\zeta=m(y+\rmi x/\kappa )\).

Thus, we obtain the induced covariant transform \(\oper{W}_{\Phi}^{\uir{}{m}}\) from \(\FSpace{L}{2}(\Space{T}{2})\) into \(\FSpace{F}{2}(\Space{R}{2})\). \index{pre-theta transform} We call \(\oper{W}_{\Phi}^{\uir{}{m}}\) the \emph{pre-theta transform}. \index{the pre-theta space! \(F_{\Theta}(\Space{R}{2})\)

 The general properties of the covariant transform \(\oper{W}_{\Theta}^{\uir{}{m}}\) yield corresponding properties of the pre-theta transform.
\begin{cor}
	 The pre-theta transform \(\oper{W}_{\Phi}^{\uir{}{m}}:\FSpace{L}{2}(\Space{T}{2})\rightarrow \FSpace{F}{2}(\Space{R}{2})\)~\eqref{theta-transform-1}
 is a unitary intertwining operator between the lattice representation \(\uir{}{m}\) on \(\FSpace{L}{2}(\Space{T}{2})\) and the left quasi-regular representation \(\Lambda_{\myhbar}\) restricted to \(F_{\Phi}(\Space{R}{2})\).
	\end{cor}

\section{Peeling representations of \(\Space[p]{H}{1}\) and analyticity}\label{sec: Peeling}

Induced representations are a common tool to construct representations of groups. However, a representation prepared  using a generic methodology may not be particularly suited for a special situation. It often needs to be tuned  to be enriched with useful features. In this section, we demonstrate such a simple tool which produces the required enhancements needed for  various situations.

For example: since the annihilation operator \(a^{-}\) provides a useful characterisation of an irreducible component of a representation, we are interested in expressing  \(a^{-}\) in the most transparent form. A map, called here \emph{peeling}, then simplifies the corresponding annihilation operator into a linear combination of first-order derivatives only.
Therefore, the structure of the eigenvectors \(\phi_n\) (cf~\citelist{\cite{Alamer19a}*{\S~5.4}\cite{Kirillov04a}*{\S~2.6}}) forms an orthonormal basis of the initial irreducible space \(\FSpace{H}{}\) becomes more transparent.

On the other hand, if a representation is reducible, its irreducible component can be characterised as the space of null-solutions for the Lie derivative  (cf.~\eqref{pre-FSB}). In such cases, we seek to peel the irreducible component to a space of analytic functions. This  allows one to use the power of complex variable theory to study the induced representations of \(\Space[p]{H}{1}\).

\begin{defn}
Let \(G\) be a group and \(\uir{}{}\) be a representation on a space \(\FSpace{L}{2}(X,\rmd\mu(x))\) of square-integrable functions on some set \(X\) with an appropriated measure \(\rmd\mu(x)\).
\index{peeling!peeling operator \(\varepsilon_d\)}
A \emph{peeling} \(\varepsilon_d\) is an invertible operator of multiplication defined by a function \(d(x)\) on \(X\):
\begin{equation}\label{peeling operator}
	\varepsilon_d: f(x)\mapsto e^{d(x)} f(x).
\end{equation}
The operator \(\varepsilon_d\) is unitary for suitably related measures:
 \begin{equation}\label{norm-unutarry-operator}
 	\varepsilon_d: \FSpace{L}{2}(X,\rmd\mu(x))\rightarrow \FSpace{L}{2}( X, \rmd\nu(x))
 \end{equation}
 such that \(\rmd\nu(x)=e^{- 2\Re d(x)}\rmd\mu(x)\).
 \end{defn}

In this paper, all considered peelings use smooth \(d(x)\) on a
domain \(X\) in a Euclidean space.
We will discuss the choice of \(d(x)\) for the pre-FSB, Schr\"odinger and lattice representations in Subsections~\ref{exm:peeling-rep-center-FSB},~\ref{peeling shrodinger} and~\ref{sub: peeling-Lattice}, respectively.

\subsection{Peeling the (pre-)FSB representation}\label{exm:peeling-rep-center-FSB}
Let \(\Lambda_{\myhbar}\) be the pre-FSB representation~\eqref{eq:H1-pol-rep-plane}, which acts irreducibly on the pre-FSB space \(\FSpace{F}{\phi_{\myhbar\kappa}}\)~\eqref{pre-FSB}.
 Consider the variables \(z, \bar{z}\in\Space{C}{}\), where \(z=\sqrt{\frac{h}{2\kappa}}(x+\rmi\kappa y)\) and \(h=2\pi\myhbar>0\).
 In this subsection, we peel the representation \(\Lambda_{\myhbar}\) into the corresponding one \(\tilde \Lambda_{\myhbar}\) that acts
on the FSB space of analytic functions. To perform this, we look for a peeling operator satisfying the following conditions:
\begin{enumerate}
\item \label{con1peeli} The peeling defined by \(\rme^{d(z, \bar{z})}\) shall intertwine the right annihilation operator \(\linv{\kappa X+\rmi Y} =2\pi\myhbar x+(\kappa \partial_{x}+\rmi \partial_{y})\) and the Cauchy--Riemann operators \(\partial_{\bar z}=\kappa \partial_{x}+\rmi\partial_{y}\):
  \begin{equation}
    \label{intertwining with lie derivative}
	\partial_{\bar{z}}~\rme^{d(z, \bar{z})}f(z,\bar {z})=\rme^{d(z, \bar{z})}\linv{\kappa X+\rmi Y}f(z, \bar{z}).
\end{equation}
A simple differential equation for~\eqref{intertwining with lie derivative} implies that
\begin{equation}
  \label{eq:FSB-peeling-first}
  d(z,\bar {z})=\tilde\psi(z) + \frac{1}{4}(z+\bar {z})^2\,,
\end{equation}
 where \(\tilde\psi\) is an arbitrary smooth function of \(z\) alone.
\item \label{con2peeli} Let \(a^-_{\Lambda_{\myhbar}}=\rmd\Lambda_{\myhbar}^{\kappa X-\rmi Y}=2\pi\rmi\myhbar\kappa y-(\kappa \partial_{x}-\rmi\partial_{y})\) be the left annihilation operator of \(\Lambda_{\myhbar}\)~\eqref{eq:H1-pol-rep-plane}. The same peeling  shall intertwine \(a^-_{\Lambda_{\myhbar}}\) with (a multiple of) the complex derivative \(\partial_{z}=(\kappa \partial_{x}-\rmi\partial_{y})\). This fixes \(\tilde\psi(z)=-\frac{1}{2}z^2-c\), \(c\in\Space{C}{}\) in~\eqref{eq:FSB-peeling-first}. Thus, the peeling operator becomes
  \begin{equation}
    \label{unique-FSB-peeling}
 \begin{split}
   \varepsilon_d\cdot I= \rme^{d(z, \bar z)}\cdot I&=\rme^{\frac{h}{4\kappa}(x^2+\kappa^2 y^2-2\rmi\kappa x y)-c }\cdot I\\
   &=\rme^{\frac{1}{4}(\bar{z}^2-z^2+2z\bar{z})-c}\cdot I.
 \end{split}
 \end{equation}
\end{enumerate}
Let \(c_0\in\Space{C}{}\) and  \(c_0\neq0\). There is a special vacuum of the representation \(\Lambda_\myhbar\) annihilated by both
 \(a^-_{\Lambda_\myhbar}\) and \(\linv{\kappa X+\rmi Y}\), given as follows:
\begin{equation}\label{vacuum-FSB-lie1}
  \begin{split}
    \phi_0(z,\bar {z}) &=c_0\,\rme^{-\frac{1}{2}\bar {z}^2+\frac{1}{4} (z-\bar {z})^2+c}\\&=c_0\,\rme^{\frac{1}{4}(z^2-\bar {z}^2-2z\bar {z})+c}.
  \end{split}
\end{equation}
The consequence of the conditions~\eqref{con1peeli} and~\eqref{con2peeli} is that the peeling maps the vacuum \(\phi_{0}\), which is killed by both the left and right annihilation operators to the function identically equal to \(c_0\), which is killed by both \(\partial_{z}\) and \(\partial_{\bar{z}}\). The peeled representation \(\tilde{\Lambda}_{\myhbar}\) is:
\begin{equation}
  \label{FSB-Rep}
  \begin{split}
    [\tilde{\Lambda}_{\myhbar}(s,z)F] (z')& \colonequals \rme^{\frac{1}{4}(\bar{z'}^2-z'^2+2z'\bar{z'})}\Lambda_{\myhbar}(s,z)\rme^{-\frac{1}{4}(\bar{z'}^2-z'^2+2z'\bar{z'})}F(z')
    \\
    &=\rme^{h\rmi s
      +\frac{1}{4}(\bar z^2-z^2-2z\bar z)+\bar z z' }F(z'-z
    ,
  \end{split}
\end{equation}
which is called the \emph{FSB representation}.
The composition of the peeling~\eqref{unique-FSB-peeling} with the covariant transform~\eqref{eq:prefsb-transform} is:
 \begin{equation}\label{FSB-transform}
   \begin{split}
     F(z)&=\left(\frac{\myhbar}{\kappa}\right)^{1/2}\rme^{\frac{h}{4\kappa}(x^2+\kappa^2 y^2-2\rmi\kappa xy)-c }\int_{\Space{R}{}} f(t)\, \rme^{2\pi\rmi\myhbar t y}\, \rme^{-\frac{\pi \myhbar}{\kappa} (t-x)^2}\,\rmd t\\
     &=\left(\frac{\myhbar}{\kappa}\right)^{1/2}\int_{\Space{R}{}} f(t)\, \rme^{-\frac{h}{2\kappa} t^2+\sqrt{\frac{2h}{\kappa}}~ t z-\frac{1}{2}z^2}\,\rmd t,
	\end{split}
\end{equation}
where \(F(z)\) is an analytic function of \(z=\sqrt{\frac{h}{2\kappa}}(x+\rmi \kappa y)\).
Indeed, by Prop.~\ref{co:cauchy-riemann-integ} and the intertwining property~\eqref{intertwining with lie derivative}, the function \(F(z)\)~\eqref{FSB-transform}
satisfies
\begin{equation*}
  \partial_{\bar{z}} F(z) = (\kappa\partial_x+\rmi\partial_y)F(x,y)=0\,,
\end{equation*}
which is essentially the Cauchy-Riemann equation.
The integral~\eqref{FSB-transform} is known as the \emph{FSB transform}.
\index{FSB space \(\FSpace[\myhbar]{F}{2}\)}
 The image \(\FSpace[\myhbar]{F}{2}\) of the FSB transforms is called the FSB space.
 It is a closed subspace of
 \begin{equation}\label{eq: (FSB)space}
   \begin{split}
     \FSpace{L}{2}(\Space{R}{2},\rme^{-\frac{h}{2\kappa}(x^2+\kappa^2 y^2)+2 c}\rmd x\,\rmd y)=\FSpace{L}{2}(\Space{C}{},\rme^{-|z|^2+2 c}\rmd z\,\rmd\bar z)
   \end{split}
 \end{equation}
 consisting of the analytic functions.
Note that often, only the values \(\myhbar=1\), \(\kappa=1\) and \(c=0\) are used
\cite{Vasilevski99}.

Similar to the pre-FSB transform, we calculate the composition of the pre-theta transform \(\tilde f\)~\eqref{theta-transform-1} and the peeling~\eqref{unique-FSB-peeling}, \(h=2\pi m\), as follows:
 \begin{equation}\label{FSB-transform-theta}
 \begin{split}
   \tilde F(z)&=\rme^{\frac{h}{4\kappa}(x^2+\kappa^2 y^2-2\rmi\kappa xy)-c }\,\tilde f(x,y)\\
   &=\int_{\Space{T}{2}}f(u,v)~\rme^{-\frac{\pi m}{\kappa}(u^2+2\rmi\kappa uv)-\frac{1}{2} z^2+2\sqrt{\frac{\pi m}{\kappa}} u z-c}\,\bar\Theta_{m\kappa}(-\rmi (\bar{z'}-\bar{z}),\rmi)\,\rmd u\,\rmd v \, ,
 \end{split}
\end{equation}
where \(z=\sqrt{\frac{h}{2\kappa}}(x+\rmi \kappa y)\in\Space{C}{}\) as before.

 We call \(\tilde F(z)\) the \emph{theta transform} of \(f(u,v)\).
As with the FSB transform, the intertwining property~\eqref{intertwining with lie derivative} implies that the image of the theta transform consists of analytic functions, which can be found in many references~\citelist{\cite{Folland89}\cite{Neretin11a}\cite{Berndt07a}}.

\subsection{Peeling the Schr\"odinger representation}\label{peeling shrodinger}
In this subsection, we are performing less common peeling  the Schr\"odinger representation \(\uir{}{\myhbar}\)~\eqref{eq:H1-schroedinger-rep}, so that the corresponding annihilation operator will  only be the derivative \(\partial_t\).  For a representation space realised as a function on a set, we have the following simple result.
 \begin{lem}
   \label{le:peeling-vacuum}
   Let the annihilation operator have the form \(a^{-}= M+D\) on \(\FSpace{L}{2}(X)\), where \(M\) is a multiplication operator and \(D\) is a derivative satisfying the Leibniz rule. For the vacuum vector \(\phi(x)\), such that \(a^{-} \phi(x)= 0 \), the peeling \(f(x) \mapsto \phi(x) f(x)\) intertwines the annihilation operator \(a^{-}\) with the derivative \(D\).
 \end{lem}

Recall  the annihilation operator
\(a^-_{\uir{}{\myhbar}}=\rmd\uir{\kappa X-\rmi Y}{\myhbar}=-2\pi\myhbar t-\kappa\partial_t\)~\eqref{annihilation-schrodinger} for the Schr\"odinger representation.
By Lem~\ref{le:peeling-vacuum}, a function \(\varepsilon_d\) which intertwines \(a^-_{\uir{}{\myhbar}}\) with the plain derivative \(\kappa \partial_t\) shall satisfy the vacuum condition: \(a^-_{\uir{}{\myhbar}} \varepsilon_d^{-1} = 0\).  Thus, the peeling defined by \(\varepsilon_d(t)=c\,\rme^{\frac{\pi\myhbar t^2}{\kappa}}\), \(c\in \Space{C}{}\) is unitary:
 \begin{equation}\label{schro-peeled-space}
 	\varepsilon_d: \FSpace{L}{2}(\Space{R}{},\rmd t)\longrightarrow \FSpace{L}{2}(\Space{R}{},\rme^{-2\frac{\pi\myhbar}{\kappa} t^2}\rmd t).
 \end{equation}
 The composition of the Schr\"odinger representation with the peeling acting on \(\FSpace{L}{2}(\Space{R}{},\rme^{-2\frac{\pi\myhbar}{\kappa} t^2}\rmd t)\) is:
  \begin{equation}\label{schr-peeled-rep}
    \begin{split}
      [\tilde\rho_{\myhbar}(s,x,y)F](t) & \colonequals
\rme^{\frac{\pi\myhbar}{\kappa} t^2}\rme^{2\pi \rmi \myhbar
  (s-t y)}\rme^{-\frac{\pi\myhbar}{\kappa} (t-x)^2}F(t-x)\\
&=\rme^{2\pi \rmi \myhbar s}\rme^{-\frac{\pi\myhbar}{\kappa} (x^2-2t(x-\rmi\kappa y))}F(t-x)\,.
 \end{split}
 \end{equation}
Consequently, the corresponding derived representations of the Lie algebra \(\algebra{h}_1\) are
 \begin{eqnarray}
 	\rmd\tilde\rho_{\myhbar}^{ X}=2\pi\myhbar t-\partial_t\quad \text{and} \quad\rmd\tilde\rho_{\myhbar}^{Y}=-2\pi\rmi\myhbar t,
 \end{eqnarray}
and the annihilation operator \(a^-_{\tilde\rho_{\myhbar}}\) is
\begin{equation}\label{annih-cor-schr}
	a^-_{\tilde\rho_{\myhbar}}=\rmd\tilde\rho_{\myhbar}^{\kappa X-\rmi Y}=-\partial_t,
\end{equation}
which annihilates the vacuum  \(\tilde\phi(t)=c\).
A notable consequence is that \(\varepsilon_d\) transforms the Hermite functions \(H_n(t)\rme^{-\frac{\pi\myhbar}{\kappa} t^2}\in\FSpace{L}{2}(\Space{R}{})\) to the corresponding \index{Hermite polynomials} \emph{Hermite polynomials} \(H_n(t) \in \FSpace{L}{2}(\Space{R}{},\rme^{-2\frac{\pi\myhbar}{\kappa} t^2}\rmd t)\).

\subsection{Peeling the lattice representation}\label{sub: peeling-Lattice}

The purpose of peeling the lattice representation \(\uir{}{m}\)~\eqref{induced-smooth-rep-from-H_d} seems to be difficult to formulate while staying within the framework of the Heisenberg group itself. However, the lattice representation is better understood if it is extended to the Schr\"odinger group (aka the Jacobi group~\cite{Berndt07a}*{ch.~9})---the semi-direct product of \(\Space[p]{H}{1}\) with the group \(\SL\) acting on  \(\Space[p]{H}{1}\) through symplectic automorphism~\cite{Folland89}*{\S~1.2}. The  representation of the Schr\"odinger group can then be peeled in a way that its vacuum will be a null-solution of the heat equation.

Since an accurate treatment of the Schr\"odinger group is beyond the scoop of the present paper, we simply provide the form of the required peeling, cf.~\eqref{calculating-theta-vacuum}:
\begin{equation}\label{lattice-peeling}
  \begin{split}
    \varepsilon_d\cdot I
    &= \rme^{-\pi\kappa (3\omega^2-\bar\omega^2-2\omega\bar{\omega})/4m}\cdot I\,, \quad
    \text{ where } \omega = \frac{m}{\kappa}(\kappa v+\rmi u).
  \end{split}
\end{equation}
Then, \(\varepsilon_d \) is a unitary operator:
 \begin{equation}\label{norm-discrete-unutarry-operator}
 \begin{split}
   \varepsilon_d: \FSpace{L}{2}(\Space{T}{2}, \rmd\omega\,\rmd\bar{\omega}) &\longrightarrow \FSpace{\hat L}{2}(\Space{T}{2}, \rme^{\pi\kappa(\omega- \bar{\omega})^2/2m}\,\rmd\omega\,\rmd\bar{\omega})
 \end{split}
 \end{equation}
Note that \(\pi\kappa(\omega -\bar{\omega})^2/2m=-2\Re[-\pi\kappa (3\omega^2-\bar\omega^2-2\omega\bar{\omega})/4m]= -2\pi m u^2/\kappa \).
The corresponding irreducible lattice representation \(\tilde\rho_m\) is
\begin{equation}\label{eq: discrete-rep-Theta}
  \begin{split}
    [\tilde\rho_m (s,\omega)F](\omega',\bar{\omega'})
    &=\rme^{2\pi m\rmi s}~\rme^{\pi\kappa(\frac{1}{4}(\omega -\bar{\omega})^2+\omega(\bar{\omega'}-\omega'))/m}F(\omega'-\omega, \bar{\omega'}-\bar{\omega})\,.
\end{split}
\end{equation}
Therefore, the corresponding annihilation operator is simply
\begin{equation}
	a^-_{{\tilde\rho}_{m}}= \rmd{\tilde\rho_m}^{\kappa X-\rmi Y}=2\rmi  m~ \partial_{\bar{\omega}},
\end{equation}
 which annihilates the theta function \(\tilde\Phi=:c\, \Theta_{m\kappa}(\omega,\frac{\rmi m}{\kappa})\).
Therefore, the peeling maps the vacuum vector \(\Phi_{m\kappa}\) from \S~\ref{integration-transform-theta} to the theta function:
 \begin{equation}\label{vacuum-peeling}
 \begin{split}
   \varepsilon_d: \Phi(\omega,\bar \omega) &\mapsto
   c~\Theta_{m\kappa}(\omega,\frac{\rmi m}{\kappa})\,.
 \end{split}
 \end{equation}

\section{A Contravariant transform on \(\Space[p]{H}{1}\)}
\label{Contravariant Transform}

The goal of  this section is to introduce the \emph{contravariant transform} \(\oper{M}_\psi\), which is the \emph{adjoint} of the covariant transform \(\oper{W}_\phi\) \citelist{\cite{AliAntGaz14a}*{\S~8.1}\cite{Kisil98a}*{\S~2}}.

\subsection{A Contravariant transform for induced representations}
\label{sec:contr-transf-induc}

Let \(H\) be a closed subgroup of \(\Space[p]{H}{1}\) and \(X=\Space[p]{H}{1}/H\) be the respective homogeneous space, which is a subset of Euclidean space with the respective Lebesgue measure.\index{reconstruction vector}
 Let \(\uir{}{}\) be a representation of the Heisenberg group \(\Space[p]{H}{1}\) on the vector space \(\FSpace{V}{}\).
 \begin{defn}
   \label{de:contravariant-tran}
   For a \emph{reconstruction vector} \(\psi\in \FSpace{V}{}\), the \emph{contravariant transform} \(\oper{M}_\psi\)
   is a map \(\oper{M}_\psi^{\uir{}{}}: \FSpace{L}{1}(X) \rightarrow \FSpace{V}{}\) given by
   \begin{equation}\label{contravariant transform}
     \begin{split}
       \oper{M}_\psi^{\uir{}{}}: ~\tilde \nu \mapsto \oper{M}_\psi(\tilde \nu )
       &=\int_{X}\tilde \nu (x)\,\uir{}{}(\map{s}(x))\, \psi\,\rmd\mu(x
       ,
     \end{split}
   \end{equation}
   where
   \(\map{s}\) is a
   Borel section from \(X=\Space[p]{H}{1}/H\) to
   \(\Space[p]{H}{1}\).
 \end{defn}

It is naturally to request that the contravariant transform \(\oper{M}_\psi^{\uir{}{}}\) shall intertwine the induced representation  \(\uir{}{\chi}\)~\eqref{eq:def-ind} and the representation \(\uir{}{}\):
\begin{equation}
  \label{eq:contravariant-intertwining}
  \oper{M}_\psi^{\uir{}{}} \circ \uir{}{\chi}(g) = \uir{}{}(g) \circ   \oper{M}_\psi^{\uir{}{}}, \qquad \text{ for all } g \in \Space[p]{H}{1}\,.
\end{equation}
The left-hand side explicitly is:
\begin{align}
  \nonumber
  [\oper{M}_\psi^{\uir{}{}} \circ \uir{}{\chi}(g) \tilde{\nu}]
  &= \int_{X} \bar{\chi}(\map{r}(g^{-1} * \map{s}(x)))\,
    \tilde{\nu} (g^{-1}\cdot x) \,\uir{}{}(\map{s}(x))\, \psi\,\rmd\mu(x)\\
  \nonumber
    &= \int_{X}
    \tilde{\nu} (y) \, \bar{\chi}(\map{r}(g * \map{s}(y)))\,\uir{}{}\left(g*\map{s}(y)*(\map{r}(g*\map{s}(y)))^{-1}\right)\, \psi\,\rmd\mu(x)\\
  \nonumber
    &= \int_{X}
    \tilde{\nu} (y) \, \bar{\chi}(\map{r}(g * \map{s}(y)))\,\uir{}{}\left(\map{s}(\map{p}(g*\map{s}(y)))\right)\, \psi\,\rmd\mu(x)\,,
\end{align}
where  \(y = g^{-1}\cdot x\) for  \(x\), \(y\in X\) and  \(g\in G\). We  used the following identities:
\begin{align}
  \label{eq:s-map-for-g-action}
  \map{s}(x) & = g*\map{s}(y)*(\map{r}(g*\map{s}(y)))^{-1},\\
  \label{eq:r-map-for-g-action}
  \map{r}(g^{-1}*\map{s}(x))& = \map{r}(g*\map{s}(y)).
\end{align}
The right-hand side of~\eqref{eq:contravariant-intertwining} is
\begin{displaymath}
[\uir{}{}(g) \circ   \oper{M}_\psi^{\uir{}{}} \tilde{\nu}] = \int_{X}\tilde \nu (y)\,\uir{}{}(g*\map{s}(y))\, \psi\,\rmd\mu(y)  \,.
\end{displaymath}
If the intertwining property holds for every function \(\tilde{\nu}\), then the following condition for the reconstructing vector \(\psi\) is required:
\begin{equation}
  \label{eq:condition-intertwining}
  \bar{\chi}(\map{r}(g))\,\uir{}{}\left(\map{s}(\map{p}(g))\right) \psi=  \uir{}{}(g) \psi\,, \qquad \text{ for all } g \in G\,.
\end{equation}

The abstract framework of the contravariant transform is well known for a unitary irreducible representation \(\uir{}{}\) in a Hilbert space \(\FSpace{V}{}\) (see for example~\cite{AliAntGaz14a}*{\S~8.1}).

For suitable fiducial \(\phi\) and reconstruction \(\psi\in \FSpace{V}{}\) vectors, the contravariant transform \(\oper{M}_\psi\) and the covariant transform \(\oper{W}_\phi\)~\eqref{eq:induce-wavelet-transform-1} are adjoints.

There are several extensions of the constructions for a strongly continuous representation \(\uir{}{}\) in a Banach space \(\FSpace{V}{}\)~\citelist{\cite{FeichGroech89a} \cite{FeichGroech89b} \cite{Kisil98a} \cite{Kisil13a} \cite{Albargi13a}}. In this paper we omit generalities and use specialised techniques for \(\Space[p]{H}{1}\) ~\citelist{\cite{Kisil98a} \cite{Kisil13a} \cite{Albargi13a}}.
We use  \(\oper{W}^*: B^* \rightarrow W^*(X)\) and \(\oper{M}^*: W^*(X) \rightarrow B^*\) to denote the adjoint operators to \(\oper{W}\) and \(\oper{M}\), respectively. This results  in  the following identity:
\begin{equation}
	\scalar{\oper{M}v}{\oper{M}^*l}_{W(X)}=\scalar{v}{l}_{B},\quad v\in B,~l\in B^*.
\end{equation}

The contravariant transform construction is particularly simple for maps acting from the pre-FSB space as the next two examples show.

\begin{example}[The inverse of the (pre-)FSB transform]
For the Schr\"odinger representation \(\uir{}{\myhbar}\)~\eqref{eq:H1-schroedinger-rep}, the intertwining condition~\eqref{eq:condition-intertwining} is trivially satisfied by an arbitrary reconstructing vector \(\psi\). From the sesqui-unitary property~\eqref{generaL-sesqui-unitary}, it follows that \(\oper{M}_{\psi}\circ \oper{W}_\phi = I\) for vectors \(\psi\) and \(\phi\) such that \(\scalar{\phi}{\psi}=1\). In particular, for the Gaussian \(\psi_{\myhbar\kappa}(t) =2^{1/4} \rme^{-\frac{\pi\myhbar}{\kappa} t^2}\)  as a reconstruction vector,  the contravariant transform is:
\begin{equation}
  \begin{split}
    [\oper{M}_\psi^{\uir{}{\myhbar}}f](t)&= 2^{1/4} \int_{\Space{R}{2}}f(x,y)~\rme^{-2\pi \rmi \myhbar ty}~\rme^{-\frac{\pi\myhbar}{\kappa} (t-x)^2}~\rmd x\,\rmd y \\ &=2^{1/4}\rme^{-\frac{\pi\myhbar}{\kappa}t^2} \int_{\Space{R}{2}}f(x,y)~\rme^{-\frac{\pi\myhbar}{\kappa} (x^2-2t(x-\rmi \kappa y))}~\rmd x\,\rmd y.
  \end{split}
\end{equation}
This is known as the \emph{inverse of the pre-FSB transform}~\cite{Neretin11a}*{\S~4.2}.
\end{example}

\begin{example}[The inverse of the (pre-)theta transform]
  A contravariant transform \( 	\oper{M}_{\Theta}^{\uir{}{m}}:\FSpace{L}{2}(\Space{R}{2})\rightarrow \FSpace{L}{2}(\Space{T}{2})\) is similar to the previous case: there are no restrictions from the intertwining condition~\eqref{eq:condition-intertwining}, and the only requirement for a reconstruction vector \(\psi\) is \(\scalar{\phi}{\psi}=1\).
In particular, if we set the reconstruction vector \(\psi\) by the lattice representation's vacuum~\eqref{calculating-theta-vacuum}
 the integral transformation \(\oper{M}_\psi^{\uir{}{m}}\) becomes:
\begin{equation}\label{inver-theta-tra}
\begin{split}
  \oper{M}_{\psi_{\myhbar\kappa}}^{\uir{}{m}}(f)&= \rme^{-\frac{\pi m}{\kappa}u^2-2\rmi\kappa u v} \int_{\Space{T}{2}}f(x,y)\,\rme^{-\frac{\pi m}{\kappa}(x^2-2 u (x- \rmi\kappa y))}\\
  &\qquad \qquad
  \times \Theta_{m\kappa}\left(\sqrt{\frac{h}{2\kappa}}(\kappa (v-y)+\rmi( u-x),\rmi\right)\,\rmd x\,\rmd y.
\end{split}
 	\end{equation}
Thus, we obtain the inverse operator of \(\oper{W}_{\Theta}^{\uir{}{m}}\)~\eqref{theta-transform-1}. We call the transformation \(\oper{M}_{\Theta}^{\uir{}{m}}\)~\eqref{inver-theta-tra} the inverse of the pre-theta transform.
\end{example}
\begin{example}[The Fourier transform]
  We can look for a contravariant transform which intertwines the two forms~\eqref{eq:H1-schroedinger-rep} and~\eqref{eq:H1-schroedinger-rep-alt} and will be an inverse of the covariant transform (the Fourier transform) from  Example~\ref{ex:covariant-Fourier}. Using maps~\eqref{eq:maps-p-s-r-H_y-prime} and  representation~\eqref{eq:H1-schroedinger-rep}, we obtain the form  of the compatibility condition~\eqref{eq:condition-intertwining}:
  \begin{displaymath}
    \rme^{2\pi \rmi \myhbar  s} \rme^{-2\pi \rmi \myhbar  ty}\, \psi(t)
    = \rme^{2\pi \rmi \myhbar  (s-ty)}\, \psi(t-x).
  \end{displaymath}
  which again delivers the solution \(\psi(\lambda)\equiv 1\). The respective contravariant transform is, as expected, the Fourier transform:
  \begin{displaymath}
    [\oper{M}_{1}^{\uir{}{\myhbar}} f](t)
    = \int_{\Space{R}{}} f(\lambda)    \rme^{-2\pi \rmi \myhbar t \lambda}\, \rmd \lambda \,.
  \end{displaymath}
\end{example}
A bit more care is required in the next case.

\subsection{The inverse of the Zak transform}
\label{inver-zak-contravariant-schro}

In Subsection~\ref{exp:calculating-Zak}, we derived the co-Zak transform \({\oper{Z}}:\FSpace{L}{2}(\Space{R}{})\rightarrow \FSpace{L}{2}(\Space{T}{2})\)~\eqref{eq:covariant-Zak1} through the induced covariant transform \(\oper{W}_{\phi_0}^{\uir{}{\myhbar}}\).
 Now, we calculate its inverse using the contravariant transform.
The intertwining property~\eqref{eq:condition-intertwining} for the map \(\map{r}(s,x,y)=(s-\{x\}[y],[x],[y])\) from~\eqref{eq:lattice-map-s-r} requires:
\begin{align}
  \nonumber
    \bar{\chi}(s-\{x\}[y],[x],[y])\,\uir{}{}\left((s,x,y)*(s-\{x\}[y],[x],[y])^{-1}\right) \psi&=  \uir{}{}(s,x,y) \psi \, ; \\
\nonumber
  \bar{\chi}(s-\{x\}[y],[x],[y])\,\uir{}{}\left((s,x,y)*(-s+x[y],-[x],-[y])\right) \psi&=  \uir{}{}(s,x,y) \psi \, ; \\
\nonumber
  \bar{\chi}(s-\{x\}[y],[x],[y])\,\uir{}{}(0,\{x\},\{y\}) \psi
                                                                                &=  \uir{}{}(s,x,y) \psi \, ; \\
  \nonumber
    \rme^{2\pi \rmi m (s-\{x\}[y])}\,
    \rme^{2\pi \rmi m (-t\{y\})}\, \psi(t-\{x\})
                                                                                &=  \rme^{2\pi \rmi m (s-ty)}\, \psi(t-x) \, ; \\
  \label{left}
        \rme^{2\pi \rmi m ((t-\{x\})[y])}\, \psi(t-\{x\})
    &=  \, \psi(t-x)\, .
\end{align}
Choosing \(y=1\) and any \(0<x<1\), we conclude that \(\psi_0\) is supported at \(\{0\}\). Further analysis shows that the reconstruction vector \(\psi_0\) satisfying the condition~\eqref{left} is the Dirac delta distribution \(\delta(t)\) (up to a multiple).

 Let \(\tilde x=(u,v)\in X=\Space{T}{2}=\Space[p]{H}{1}/H_d\), where \(H_d\) is the non-commutative subgroup~\eqref{non-commuta}. For the section map
 \(\map{s}(u,v) = (0,u,v)\)~\eqref{eq:lattice-map-s-r} and \(g\in\FSpace{L}{2}(\Space{T}{2})\), the contravariant transform becomes:
 \begin{equation}
   \label{eq:invers-zak1}
   \begin{split}
     \oper{M}_{\psi_0}^{\uir{}{\myhbar }}: g \mapsto & \int_{\Space{T}{}}\int_{\Space{T}{}}g(u,v)\,\uir{}{\myhbar}(\map{s}(\tilde x) )\,\psi_0(t)\,\rmd u\,\rmd v\\&=\int_{0}^1\int_{0}^1g(u,v)\,\uir{}{\myhbar}(0,u,v)\,\psi_0(t)\,\rmd u~\rmd v\\&=\int_{0}^1\int_{0}^1g(u,v) \,\rme^{-2\pi\rmi m t v}\delta(t-u)\,\rmd u\,\rmd v\\&=\int_{0}^1g(t,v) ~\rme^{-2\pi\rmi m t v}\,\rmd v\\&= \int_0^1 \tilde{g}(t,v) ~\rmd v.
   \end{split}
 \end{equation}
 Since \(g(t,v)\) is contained in the space \( \FSpace{L}{2}(\Space{T}{2})\) of square-integrable functions that are periodic in \(t\) and quasi-periodic in \(v\), multiplying \(g(t,v)\) by \(\rme^{-2\pi\rmi m t v}\) produces a function that has the same double quasi-periodicity property of \(g(t,v)\) but in the opposite way. In other words,
 	 \(\tilde{g}(t,v)=g(t,v)\cdot\rme^{-2\pi\rmi m t v}\in\FSpace{\tilde L}{2}(\Space{T}{2})\) is quasi-periodic in \(t\) and periodic in \(v\) and square-integrable.
 	Moreover, since \(t\in \Space{R}{}\approx [0,1]\times\Space{Z}{}\), then \(t=x+n\), for some \(x\in [0,1]\) and \(n\in\Space{Z}{}\).
 	Therefore, for \(t=x+n\),~\eqref{eq:invers-zak1} becomes
 	\begin{equation}\label{eq:invers-zak}
 	\begin{split}
 		\int_{0}^1\tilde{g}(x+n,v)~\rmd v &=\int_{0}^1\tilde{g}(x,v) ~\rme^{-2\pi\rmi m n v}~\rmd v\\&=[\oper{M}_{\psi_0}^{\uir{}{\myhbar }} g](x)=[\oper{M}_{\psi_0}^{\uir{}{\myhbar }} (\rme^{2\pi\rmi m x v}~\tilde g)](x).
 		\end{split}
 	\end{equation}
 Thus, \(\oper{M}_{\psi_0}^{\uir{}{\myhbar }}\) is the inverse of the induced covariant transform \(\oper{W}_{\phi_0}^{\uir{}{\myhbar }}\)~\eqref{eq:covariant-Zak1} from \( \FSpace{L}{2}(\Space{T}{2})\) into \( \FSpace{L}{2}(\Space{R}{})\).
 We now provide the standard definition of the inverse of the Zak transform.
 \begin{defn}\label{the inverse of the Zak transform}\cite{Neretin11a}*{\S~8.1}
 		Let \(\FSpace{\tilde L}{2}(\Space{T}{2})\) be a space of square-integrable functions \(\tilde g(x,v)\) that are quasi-periodic in \(x\) and periodic in \(v\). Let \(\tilde g= \oper{\tilde Z}f\) be the Zak transform of \(f\in \oper{S}(\Space{R}{})\subset \FSpace{L}{2}(\Space{R}{})\). The function \(f\) can be reconstructed using the following formula:
 	\begin{equation}\label{inversion of the Zak}
 	 	\begin{split}
                  \oper{\tilde Z}^{-1}&: \FSpace{\tilde L}{2}(\Space{T}{2})\rightarrow \FSpace{L}{2}(\Space{R}{})\\
                  &\quad [\oper{\tilde Z}^{-1}\tilde g](x)=\int_{\Space{T}{}}\tilde g(x,v)\rme^{-2\pi\rmi m n v}~\rmd v,\quad
 		n\in \Space{Z}{},~m\in \Space{N}{}.
 	\end{split}
 \end{equation}
The operator \(\oper{\tilde Z}^{-1}\)	is called {\em the inverse of the Zak transform}.
 \end{defn}
 \begin{cor}

   Let \(\tilde{g}(x,v)=g(x,v)\cdot\rme^{-2\pi\rmi m x v}\) such that \(g\in\FSpace{L}{2}(\Space{T}{2})\). The contravariant transform~\eqref{eq:invers-zak},
 	 \begin{equation}\label{eq:invers-zak-1-contra}
 	 \begin{split}
 	 	[\oper{M}_{\psi_0}^{\uir{}{\myhbar }} (\rme^{2\pi\rmi m x v}~\tilde g)](x)&=\int_{0}^1\tilde{g}(x,v) ~\rme^{-2\pi \rmi m n v}~\rmd v\\&= [\oper{\tilde Z}^{-1}\tilde g](x),
 	 \end{split}
 \end{equation}
 is the inverse of the Zak transform.\index{the inverse of the co-Zak transform}
 For \(g\in\FSpace{L}{2}(\Space{T}{2})\),
one can write the inverse of the co-Zak transform as \(\oper{Z}^{-1}g=\oper{\tilde Z}^{-1}~\rme^{-2\pi \rmi m u v}g \).
 \end{cor}

\section{Conclusion}

Our work in this paper illustrates the technique which allows to obtain co- and contra-variant transforms with desired properties. For the covariant transform, the fiducial vector needs to agree  with Pop.~\ref{co:cauchy-riemann-integ}. An induced contravariant transform will be an intertwining operator with an induced representation if the reconstructing vector satisfies~\eqref{eq:condition-intertwining}. This approach is illustrated on various representations of the Heisenberg group, and produces an interpretation of the Zak transform and its inverse as induced co- and contra-variant  transforms.
Furthermore, we used  peeling operators to obtain the familiar representation spaces of analytic function space of analytic functions.

Of course, this approach is not limited to the illustrative example of the Heisenberg group and can be fruitfully applied in many other situations.

\newpage
\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


m and a analyst cover letter. You most likely already know that m and a analyst cover letter is among the most popular topics on the net nowadays. Depending on the info we acquired from google adwords, m and a analyst cover letter has incredibly search in google search engine. We believe that m and a analyst cover letter give new concepts or references for followers.
We have learned plenty of references about m and a analyst cover letter but we feel this is the greatest. I we do hope you would also accept our opinion. This picture has been published by admin and is in category tags section. Youll be able to download this image by hitting the save link or right click the picture and choose save.
We hope whatever we give to you could be useful. If you want, you are able to share this article to your companion, family, neighborhood, or you can also book mark this page.
Investment Research Analyst Cover Letter Homework Academic Writing .
Application Support Analyst Cover Letter Sample LiveCareer.
Elegant 20 Financial Analyst Cover Letter Financial Analyst Resume .
Benefits Analyst Cover Letter Sample LiveCareer.
M And A Analyst Cover Letter Fungramco Strong Cover Letters Best .
Energy Analyst Cover Letter Sample LiveCareer.
Distribution Analyst Cover Letter Oursearchworld Com .
Resume Example Chartered Financial Analyst Cover Letter Resume .
Best Oil And Gas Lease Analyst Cover Letter Simply Simple Oil And .
Cover Letter Examples Financial Analyst Into Anysearch Co.
Contract Analyst Cover Letter Sample LiveCareer.
Business Analyst Cover Letter Sample Livecareer Regarding Business .
Junior Business Analyst Cover Letter Sample Livecareer In Cover .
M Pictures Of Photo Albums Marketing Research Analyst Cover Letter .
Portfolio Analyst Cover Letter Sample LiveCareer.
Project Analyst Cover Letter Sample LiveCareer.
Reporting Analyst Cover Letter Sample LiveCareer.
Technical Support Analyst Cover Letter Sample LiveCareer.
Business Analyst Cover Letter Occupational Examples Samples Free .
Resume Example Customer Service Analyst Cover Letter Resume Cover .
Qa Analyst Cover Letter Sample LiveCareer.
M New Picture Marketing Research Analyst Cover Letter Resume Cover .
Tax Analyst Cover Letter Sample LiveCareer.
Accenture Analyst Cover Letter M And A Analyst Cover Letter Fungram .
Resume Example Marketing Business Analyst Cover Letter Resume .
Lead Business Analyst Cover Letter Lovely M And A Analyst Cover .
Art Galleries In Operations Research Analyst Cover Letter Research .
Pmo Analyst Cover Letter Sample LiveCareer.
Financial Analyst Cover Letter Analyst Cover Letter Sample Harfiah .
Free Data Analyst Cover Letter Sample Templates At Data Analyst .
Banking Business Analyst Cover Letter Samples Of Operationsr .
Teaching Assistant Application Letter Workforce Analyst Photo Album .
Materials Analyst Cover Letter 1 638 Jpg Cb 1411789000.
Business Analyst Cover Letter Samples Business Analyst Cover Letters .
Configuration Management Analyst Cover Letter Lovely M And A Analyst .
Cover Letter Example Finance Resume Cover Letter Examples Financial .
Workforce Management Analyst Cover Letter Simply Simple Workforce .
Gis Analyst Cover Letter Sample Collection Of Solutions Sample Cover .
Financial Analyst Cover Letter Template Images Letter Format .
Sample Cover Letter Analyst Financial Analyst Cover Letter Sample .
Private Equity Analyst Cover Letter Sarahepps Com .
Program Analyst Cover Letter Sample LiveCareer.
4 Research Assistant Cover Letter Data Analyst Resumes Cover Letter .
Operational Risk Analyst Cover Letter Data Analyst Cover Letter .
Cover Letter For Research Analyst Financial Analyst Cover Letter .
Supply Chain Analyst Cover Letter Sample LiveCareer.
Resume All Source Intelligence Analyst Cover Letter Best .
Resume Example Financial Planning Analyst Cover Letter Resume .
Cover Letter Privateity Wso Introduction Application Associate .
Cover Letter Sample For Job Finance New Credit Analyst Cover Letter .


text:
PRESS RELEASE: Health First Europe launches European Parliamentary Interest Group on Innovation in Health and Social Care
Press release, / 12 November 2014
Brussels, 12 November 2014 – Health First Europe today announces the launch of a new Parliamentary Interest Group to facilitate greater focus on improving patient access to innovation in health and social care. Together with Members of the European Parliament Marian Harkin (ALDE, IE), Karin Kadenbach (S&D, Austria) and Dr. Kay Swinburne (ECR, UK), the Interest Group will address key policy objectives around research, health promotion and public health at EU level.
Ensuring that public health is a Parliamentary priority in the next term, the Interest Group will focus its activities on prioritising innovation on the EU health and social care agenda, raising awareness of patient-centric solutions for sustainable, accessible and resilient health systems, and seeking broad support from MEPs for implementing innovative solutions for patients and healthcare professionals.
"The Parliamentary Interest Group is an important vehicle not only to support MEPs in their work on public health, but to help bring forward concrete ideas and solutions for the many challenges we face as citizens when it comes to accessing high quality health and social care." said Health First Europe's Honorary President John Bowis, "We will partner with the Co-Chairs, interested MEPs and other stakeholders to develop and implement material solutions that facilitate patient-centric care across the continuum – from prevention, to diagnosis, monitoring, treatment and cure."
The Interest Group was launched on the occasion of the 10th Anniversary of Health First Europe, bringing together officials from the European Institutions and health stakeholders to celebrate the past work of the association on a variety of key health issues. The first meeting of the Interest Group will take place in 2015.
For more information about the Interest Group, click here.
info@healthfirsteurope.org

redpajama_set_name: RedPajamaCommonCrawl


We are a group of religious persons committed to working together for the common needs of the whole community of life. Centered in our respective faiths, we wish to manifest the wisdom of the world's spiritual traditions -- our common human heritage. The Peace Council will build bridges among all beings through the message of non-violence, compassion, human rights, and universal responsibility, individual and collective.
We acknowledge our many failures and the frequent abuse of religion, but we celebrate its power to heal, to educate, to enlighten, and to transform.
We affirm that there exists within human beings the capacity for good and the spirit of peace.
Our goal is to nurture peace and good will among human beings and to preserve the integrity of the natural environment on which all life depends.
The commitment to foster cultures and practices of non-violence and reverence for all life.
The commitment to foster cultures and practices of shared responsibility and just society.
The commitment to foster cultures and practices of mutuality in the face of diversity, through right relationships and integrity.
The commitment to foster cultures and practices of partnership and equal opportunity between women and men, between the generations, and among the world's peoples and cultures.
The commitment to foster cultures and practices of ecological interdependence and sustainable relationships between humans, Earth, and all living beings.
Did he ask regarding furniture depot? This Warehouse furniture is Nathan's popular. . Sign compatibility reports, pisces cancer. Compatibility reports.


Powers/Abilities: The High-Zero was a skilled intelligence operative, and carried a pistol.
History: (Namor Annual#4 (fb) - BTS) - In his infancy, the man who would become the High-Zero was brought to Baron Wolfgang von Strucker, who raised him to become a pivotal agent of HYDRA. He was eventually rewarded with the rank of "High-Zero."
(Namor Annual#4) - The High-Zero accompanied Baron Strucker aboard an underwater HYDRA vessel as they prepared to receive the formula to Project: Blue-Face from a HYDRA facility, intending to use it to wage war on Earth's undersea races. However, the HYDRA base was destroyed by a Leviathan (offspring of Marrina and Namor) before they could transmit the information, and they had to travel to the site of its destruction to retrieve the information.
At the site of the Leviathan, they were attacked by smaller creatures created as parasites to defend it, and by the Sub-Mariner. The High-Zero advised Strucker to flee before they were destroyed, but Agent 13, one of Strucker's favorites, called him a coward. When Namor found the canister they were seeking and presented it to them, telling them to leave, the High-Zero pointed out to Strucker that they could being implementing Project: Blue-Face immediately, and Strucker agreed.
As the High-Zero patrolled the ship's corridors, he saw Agent 13 enter the room where the toxin canister was held. He confronted Agent 13, who declared that he was taking the toxin and leaving the ship. The High-Zero attacked him and tried to strangle him, but Agent 13 took the High-Zero's gun and shot him to death.
Agent 13 subsequently impersonated the High-Zero to foil Strucker's plans, and was finally unmasked as Alan Desmond, son of the toxin's creator.
Comments: Created by Karl Bollers, Nel Yomtov, Bill Wylie and Sandu Florea.


text:
Michael Kugelman, Foreign Policy
Why the India-Pakistan War Over Water Is So Dangerous
October 6, 2016 By Wilson Center Staff
The original version of this article, by Michael Kugelman, appeared on Foreign Policy.
Early on the morning of Sept. 29, according to India's Defense Ministry and military, Indian forces staged a "surgical strike" in Pakistan-administered Kashmir that targeted seven terrorist camps and killed multiple militants. Pakistan angrily denied that the daring raid took place, though it did state that two of its soldiers were killed in clashes with Indian troops along their disputed border. New Delhi's announcement of its strike plunged already tense India-Pakistan relations into deep crisis. It came 11 days after militants identified by India as members of the Pakistani terrorist group Jaish-e-Mohammed killed 18 soldiers on a military base in the town of Uri, in India-administered Kashmir.
Amid all the shrill rhetoric and saber rattling emanating from India and Pakistan in recent days – including India's home minister branding Pakistan a "terrorist state" and Pakistan's defense minister threatening to wage nuclear war on India – one subtle threat issued by India may have sounded relatively innocuous to the casual listener.
In reality, it likely filled Pakistan with fear.
On Sept. 22, India's Foreign Ministry spokesman suggested, cryptically, that New Delhi could revoke the Indus Waters Treaty (IWT). "For any such treaty to work," warned Vikas Swarup, when asked if India would cancel the agreement, "it is important for mutual trust and cooperation. It cannot be a one-sided affair."
Continue reading on Foreign Policy.
Michael Kugelman is the senior program associate for South and Southeast Asia at the Wilson Center.
Sources: Foreign Policy.
Photo Credit: A tributary to the Indus in Indian Kashmir, courtesy of flickr user sandeepachetan.com travel photography.
Topics: agriculture, China, conflict, energy, environment, environmental peacemaking, environmental security, hydropower, India, natural resources, Pakistan, population, security, South Asia, water


text:
Home Children's Theater of Madison: "The Velveteen Rabbit"
Children's Theater of Madison: "The Velveteen Rabbit"
Isthmus Pick
Overture Center-Playhouse 201 State St., Madison, Wisconsin 53703
Friday, May 8, Overture Center Playhouse, 7 pm
Are your wee ones getting spring fever? Cure it with this delightful show, adapted from Margery Williams' classic tale of a stuffed rabbit that yearns to be real. Full of fun and laughter, this play touches on love and friendship, promising to leave a smile on your face. Suitable for ages 4-plus. ALSO: Saturday and Sunday (1 & 4 pm), May 9-10. Through May 17.
CTM PRESENTS
Local and regional talent bring classic story to life
The Playhouse at Overture Center for the Arts
Based on the book by Margery Williams
Adapted by Phil Grecian
Directed by Roseann Sheridan
CTM (Children's Theater of Madison) presents The Velveteen Rabbit May 8-17 as the finale for its 49th season. True to the classic children's book by Margery Williams, this stage adaptation tells the tender story of a young boy's special companion – a velveteen rabbit. With the help of his toy friends and the magic Toy Fairy, the rabbit learns the true meaning of love, friendship and being "real" in this play filled with humor and heart.
CTM's Artistic Director, Roseann Sheridan, directs The Velveteen Rabbit. "We are so proud to bring this cherished story to our audience," says Sheridan. "The magic and joy of The Velveteen Rabbit appeals to all ages, and we hope that multi-generational families come together to enjoy this touching tale of compassion and empathy. It's a story that so many of us remember from childhood, before the movie Toy Story, but very similar to the style of Toy Story – witty, funny, touching -- with great characters and Toy Story's fun and poignancy."
CTM's casting philosophy is to cast actors in age-appropriate roles, and to anchor the show in adult talent. This method is commonly used by professional theatre for young audiences (TYA) companies. For roles where age is not critical, CTM uses a mixture of adult and child actors. The Velveteen Rabbit features fourth-generation actor Teddy Warren in the title role. Warren (24), a Milwaukee native, has appeared in many productions at First Stage Children's Theater in Milwaukee, where he was a graduate of their acclaimed Academy. Warren graduated from Southern Methodist University with a BFA in Acting and has been working professionally in the Midwest and Dallas. Following his run with CTM, Mr. Warren goes to Dallas Shakespeare Theater to play Benvolio in Romeo and Juliet.
Other adult actors new to CTM include Reggie Kellum as Bulka the Rag Puppy, Mitch Taylor as Timothy the Lion, Greg Hudson as the Skin Horse, and Dave Pausch as the Doctor. Returning adult actors Toya Robinson plays the Toy Fairy and Terry Kiss-Frank plays Nana. Child actors include: Max Vitale (11) as Alex and Lily Mefford (11) as Mouse. Seven additional children play the roles of the "real" rabbits.
CTM's Artistic Director says "I'm excited to work with this diverse and talented cast. It's really important to bring in actors in their 20's to play these kinds of roles (toys and animals) in a show like this. It anchors the production and makes characters that the youngest audience members can both look up to and relate to at the same time. It's like having a "big brother" to play with. We're really fortunate to have Teddy (Warren) with us for this show. His experience working in TYA shows is invaluable. He understands how to play with animation without being condescending. That's extremely important to me in working on shows for our very young audiences."
As part of CTM's ongoing Accessibility Initiative, CTM is providing both an ASL-interpreted performance and a sensory-friendly performance. The Friday May 8th performance at 7pm will be ASL-interpreted for the hearing impaired. A sensory-friendly performance for youth on the autism spectrum and their families will be presented on Friday, May 15 at 6:30 p.m. CTM's Accessibility Initiative is a multi-faceted effort to broaden, deepen, and diversify CTM's audiences while engaging, educating, and inspiring the next generation of theater-lovers. Efforts include ongoing partnerships with: the Autism Society of South Central Wisconsin; the Madison Children's Museum and the pilot program called Play & A Play; and no-cost afterschool classes held at the Boys & Girls Club of Dane County and the Goodman Community Center. Danielle Tolzmann, President of The Autism Society of South Central Wisconsin says, "We're so thankful for our continued partnership with the Children's Theater of Madison to offer this very special performance. Theater is magical and wonderful but might not be accessible to our families without these modifications." The sensory-friendly offering is also made possible through the support of the following sponsors: The Capital Times Kids Fund, The Evjue Foundation, the Wisconsin Arts Board, Madison Community Foundation, BMO Harris Bank, Tri-North, American Girl's Fund for Children and Endres Manufacturing Company. Tickets are available directly through the South Central Autism Society's website at autismsouthcentral.org.
Joe Varga is the show's set designer. Monica Cliff, a puppet designer whose work was seen in Midsummer Night's Dream and James and the Giant Peach, is costume and toy designer. John Pennisten is Lighting Designer and and GW Rodriguez is designing the production's sound.
The Velveteen Rabbit will be performed in The Playhouse at Overture Center for the Arts. Performances are Friday, May 8 at 7 p.m., Saturday, May 9 at 1 and 4 p.m., Sunday, May 10 at 1 and 4 p.m., Saturday, May 16 at 1 and 4 p.m., and Sunday, May 17 at 1 and 4 p.m. On Friday, May 8 at 6 p.m., there will be an opening night pre-performance party. There will be arts & crafts activities and refreshments. That evening's performance will also be ASL-interpreted for the hearing impaired. Tickets for public performances may be purchased through the CTM website (ctmtheater.org), Overture Center's website (overturecenter.com), by phone at 608-258-4141, or in person at the Overture Center Box office. Tickets are $18 for children 12 and under and $29 for adults. The suggested age for the show is 4 and up.
Location Overture Center-Playhouse 201 State St., Madison, Wisconsin 53703
Isthmus Picks Isthmus Picks
Event Categories Theater & Dance
Date & Time May 17, 2015 4:00 PM


Android: Google previewed over-the-air app installing at its I/O conference, but you can get that functionality in your Android phone right now. Install AppBrain and its Fast Web Installer, and clicking "Install" on the web will instantly install apps on your phone.
AppBrain could previously synchronize a list of your installed apps with a list of the apps you wanted on the web, as detailed in a previous post, but it still required a few clicks and a side trip into the Market to get the apps installed. With the Fast Web Installer add-on for AppBrain, you can head to AppBrain's own market, search and sort to find an app, then click "Install" and see it instantly download onto your phone.
Of course, you might be wondering how secure it is to provide AppBrain instantaneous access to install whatever it thinks you want on your phone. AppBrain's apps connect you through your Google credentials without ever actually seeing your password. AppBrain could stand to have a bit more information on their privacy page about app installing security, but they seem to be on the up-and-up about what they're collecting and why. It's up to you whether the instant-install convenience is worth trusting the developers—they have 25 utility-type apps in the Market already, though, so it doesn't quite seem like a fly-by-night scam.
One more hitch: AppBrain's instant installer only works for free applications, as you'll have to verify your payment credentials for paid apps. Beyond that, it's worked with every app we've tried out this morning.
AppBrain and the Fast Web Installer are free downloads from the Android Market. You can find QR codes to install both applications at the Androinica link.