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%\volume{228}
%\Copyrightyear{2004}
%\DOI{003-0001}
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%\seriesextraline{This is the Concrete Title of this Book\br H.E. R and S.T.C. W, Eds.}
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%\Volumeandyear{1 (2004)}
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%\DOI{003-xxxx-y}
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%\submitted{March 14, 2003}
%\received{March 16, 2000}
%\revised{June 1, 2000}
%\accepted{July 22, 2000}
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%Insert here the title, affiliations and abstract:
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\title[One approach to a generalization of white noise analysis]
      {One approach to a generalization\\ of white noise analysis}
%----------Author 1
\author{Yu. M. Berezansky}

\address{%
Institute of Mathematics\\
National Academy of Sciences of Ukraine\\
3 Tereshchenkivs'ka St.\\
0160 Kyiv\\
Ukraine}

\email{berezan@mathber.carrier.kiev.ua}

%%\thanks{This work was completed with the support of our
%%\TeX-pert.}
%----------Author 2
\author{V. A. Tesko}
\address{Institute of Mathematics\\
National Academy of Sciences of Ukraine\\
3 Tereshchenkivs'ka St.\\
0160 Kyiv\\
Ukraine}

\email{tesko@imath.kiev.ua}
%----------classification, keywords, date
\subjclass{Primary 60H40, 46F25}%; Secondary 42C05}

\keywords{Fock space, generalized functions, biunitary map,
          Schefer polynomials}

\date{October 1, 2007}
%----------additions
\dedicatory{This paper is dedicated to M. G. Krein.}
%%% ----------------------------------------------------------------------

\begin{abstract}
      In this article we review some recent developments in
      white noise analysis and its generalizations. In
      particular, we describe the main idea of the biorthogonal
      approach to a generalization of white noise analysis connected
      with the theory of hypergroups.
\end{abstract}

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\maketitle
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%\tableofcontents
\section{Introduction}

The classical white noise analysis (Gaussian white noise analysis)
it is possible to understand as a theory of generalized functions of
infinite many variables with pairing between test and generalized
functions provided by integration with respect to the Gaussian
measure. It is well known that there exist several approaches to the
construction of such theory of generalized functions: the
Berezansky-Samoilenko approach \cite{BS73} and the Hida approach
\cite{Hida75}. In the Berezansky-Samoilenko approach  spaces of test
and generalized functions are constructed as infinite tensor
products of one-dimensional spaces. The Hida approach consists in
the construction of some rigging of a Fock space with subsequent
application to the spaces of this rigging the Wiener-It\^{o}-Segal
isomorphism.

After a number of years it became clear that the Hida approach is
more convenient and in most cases all investigations in white noise
analysis and its generalizations  are based on this approach. There
exist many works dedicated to white noise analysis development:
\begin{itemize}
  \item Works deal with the investigation of
        spaces of test and generalized functions
        and operators acting in these  spaces
        using the Wiener-It\^{o}-Segal isomorphism
        and various riggings of the Fock
        space. For more information,
        see the books \cite{Hida75, BK88, HKPS93, Kuo96},
        surveys \cite{Kuo02, Kuo02a} and the references therein.
  \item Works deal with the so-called {\it Jacobi fields approach}
        to a generalization of white noise analysis.
        In these works the role of the Wiener-It\^{o}-Segal isomorphism
        is played a unitary Fourier transform which is defined by
        the Jacobi field, i.e., by some family of
        commuting selfadjoint operators that act in the Fock space and have
        a Jacobi structure.
        The theory of Jacobi fields was created by Berezansky under the
        influence of the works of M.~G.~Krein (see e.g. \cite{Krein1, Krein2})
        about Jacobi matrices. A detailed study of
        general commutative Jacobi fields in the Fock space and a
        corresponding spectral measure was carried out in the
        works by Berezansky and his collaborators, see
        e.g. \cite{B91}, \cite{B98b}--\cite{B00a}, \cite{BLLy95}--\cite{BM03},
        \cite{L95}--\cite{LU96}.
        Note that the Wiener-It\^{o}-Segal isomorphism it is
        possible to understand as the Fourier transform of a
        certain Jacobi field, so-called free field. This result
        was obtained by Koshmanenko and Samoilenko in \cite{KS75},
        see also \cite{BK88}.
  \item Works are devoted the {\it biorthogonal approach} to a generalization of
        white noise analysis. In this approach, one replaces the system
        of Hermite polynomials, that are orthogonal with respect to the
        Gaussian measure, with a certain biorthogonal system.
        The biorthogonal approach was inspired by \cite{D91}, proposed in
        \cite{AKS93} and developed in
        \cite{Us95, ADK96, BK95, BK96, Ka96, KaK99, BT03, BT04}
        (see survey \cite{BT03} for the complete
        bibliography). Note that in \cite{BK95, BK96} was first
        observed that the biorthogonal approach is deep connected
        with the theory of hypergroups.
\end{itemize}

There exists the deep analogy between the above mentioned works. In
all these works spaces of test function are constructed as image of
positive spaces from some rigging of the Fock space. But in the
first series of works is used the Wiener-It\^{o}-Segal isomorphism,
in the second series  is used a Fourier transform and in the third
series is used a certain
biunitary map.% that is constructed by using some biorthogonal system.

This survey is devoted to the biorthogonal approach to a
generalization of classical white noise analysis. In the first part
of the survey we recall the main idea of Hida approach to the
construction of classical white noise analysis. In the second part
we  give the basic idea of the biorthogonal approach. In order to
make it more simple, at first we will consider the corresponding
theory of generalized functions  for model one-dimensional case and
than briefly for infinite-dimensional. For the details and proofs we
refer the reader to the surveys \cite{BT03, BT04}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Gaussian white noise analysis}

Let us shortly recall some basic results of Gaussian white noise
analysis, for details see e.g. \cite{BK88, HKPS93}. We consider a rigging of
the real Hilbert space $L^2({\mathbb R}):=L^2({\mathbb R},dt)$,
\begin{equation*}%\label{eq.2.0}
   {\mathcal S}^{'}\supset L^2({\mathbb R}) \supset{\mathcal S},
\end{equation*}
where ${\mathcal S}$ is the Schwartz space of infinite
differentiable, rapidly decreasing function on ${\mathbb R}$,
${\mathcal S}^{\,'}$ is the Schwartz space of distributions dual
of ${\mathcal S}$ with respect to the zero space $L^2({\mathbb
R})$. We denote by $\langle\cdot\,,\cdot\rangle$ the dual pairing
between elements of ${\mathcal S}^{\,'}$ and ${\mathcal S}$
inducted by the scalar product in $L^2({\mathbb R})$, i.e., for any
$f\in L^2({\mathbb R})$ and any $\varphi\in{\mathcal S}$
$$
   \langle f,\varphi\rangle:=(f,\varphi)_{L^2({\mathbb R})}.
$$
We will
preserve this notation for tensor powers and complexifications
of spaces.

Let $\rho_{G}$   be a probability measure on the Borel
$\sigma$-algebra ${\mathcal B}(S^{\,'})$ such that
%its Fourier transforms has the form
\begin{equation}\label{eq.2.1}
   \int_{{\mathcal S}^{\,'}}e^{i\langle x,\varphi\rangle}d\rho_{G}(x)
   =e^{-\frac{1}{2}\|\varphi\|_{L^2({\mathbb R})}^{2}},\qquad
   \varphi\in {\mathcal S}.
\end{equation}
By Minlos theorem  the measure $\rho_{G}$  is  completely
characterized by (\ref{eq.2.1}). This measure $\rho_{G}$  is
called the {\it Gaussian measure}.

Note that elements $x\in S^{\,'}$ can be thought of as paths of
the derivative of Brownian motion, i.e., as white noise. More
precisely, it follows from (\ref{eq.2.1}) that
$$
   \int_{{\mathcal S}^{\,'}}\langle x,\varphi\rangle^2d\rho_{G}(x)
   =\|\varphi\|_{L^2({\mathbb R})}^{2},\qquad
   \varphi\in {\mathcal S}.
$$
Hence, extending the mapping
$$
   L^2({\mathbb R})\supset{\mathcal S}\ni\varphi\mapsto\langle\cdot,\varphi\rangle
   \in L^2({\mathcal S}^{\,'},\rho_{G})
$$
by continuity, we obtain a random variable
$\langle\cdot,f\rangle\in L^2({\mathcal S}^{\,'},\rho_{G})$ for
each $f\in L^2({\mathbb R})$. Thus, we can define the stochastic
process $\{B_t\}_{t\in{\mathbb R}}$,
\begin{equation*}
   B_t(\cdot):=
   \begin{cases}
         \langle\cdot,\varkappa_{[0,t]}\rangle, & t\geq 0, \\
         -\langle\cdot,\varkappa_{[t,0]}\rangle, & t<0
   \end{cases}
\end{equation*}
($\varkappa_\alpha$ is the indicator function of a set $\alpha$).
It is easily seen that $\{B_t\}_{t\in{\mathbb R}}$ is a version of
Brownian motion, i.e., finite-dimensional distributions of the
process $\{B_t\}_{t\in{\mathbb R}}$ coincide with those of
Brownian motion. We now informally have, for all $t\in{\mathbb
R}$,
\begin{equation*}
       B_t(x)=\int_{0}^{t}x(s)ds,\quad
       {\text{\rm so}}\;\; {\text{\rm that}}\quad \frac{d}{dt}B_{t}(x)=x(t).
\end{equation*}

The main technical tool of the construction and study of spaces of
test and generalized functions in Gaussian white noise analysis is
the {\it Wiener-It\^{o}-Segal isomorphism}
\begin{equation*}
  I_{G}:F(L^2({\mathbb R}))\to L^2({\mathcal S}^{\,'},\rho_{G})
\end{equation*}
between the symmetric  Fock space $F(L^2({\mathbb R}))$ and the
complex space $L^2({\mathcal S}^{'},\rho_{G})$. Let us recall that
the symmetric  Fock space $F(L^2({\mathbb R}))$ over $L^2({\mathbb
R})$ is defined as
\begin{equation*}
   F(L^2({\mathbb R}))
   :=\bigoplus_{n=0}^{\infty}{\mathcal F}_n(L^2({\mathbb R}))n!,
\end{equation*}
where the $n$-particle Fock space
$$
   {\mathcal F}_n(L^2({\mathbb R}))
   :=(L_{{\mathbb C}}^2({\mathbb R}))^{\mathbin{\widehat{\otimes}} n}
   \qquad ((L_{{\mathbb C}}^2({\mathbb R}))^{\mathbin{\widehat{\otimes}}
   0}:={\mathbb C})
$$
is equal to the $n$-th symmetric tensor power
$\mathbin{\widehat{\otimes}}$ of the complexification $L_{{\mathbb
C}}^2({\mathbb R})$ of the real space $L^2({\mathbb R})$
(here and subsequently, the lower index ${\mathbb
C}$ denotes complexification of a real space). Thus, for each
$f=(f_n)_{n=0}^{\infty}\in F(L^2({\mathbb R}))$,
$$
   \|f\|_{F(L^2({\mathbb R}))}^{2}=\sum_{n=0}^{\infty}
   \|f_n\|_{{\mathcal F}_n(L^2({\mathbb R}))}^{2}n!<\infty.
$$

The isomorphism $I_{G}$ is completely characterized by its
following properties:
\begin{enumerate}
  \item $I_{G}:F(L^2({\mathbb R}))\to L^2({\mathcal S}^{\,'},\rho_{G})$
        is the unitary operator.
  \item $I_{G}(f_0,0,0,\ldots)=f_0$ for all $f_0\in{\mathbb C}$.
  \item For each $n\in{\mathbb N}$ and any disjoint Borel sets
        $\alpha_1,\ldots, \alpha_n\in\mathcal{B}(\mathbb{R})$ of finite Lebesgue measure,
      \begin{equation*}
            \big(I_{G}(\underbrace{0,\ldots,0}_{n},\varkappa_{\alpha_1}\mathbin{\widehat{\otimes}}
            \cdots\mathbin{\widehat{\otimes}}\varkappa_{\alpha_n},0,0,\ldots)\big)(\cdot)
            =\langle \cdot,\varkappa_{\alpha_1}\rangle\ldots
            \langle \cdot,\varkappa_{\alpha_n}\rangle.
      \end{equation*}
\end{enumerate}
There are several equivalent ways of the construction of such
isomorphism:
\begin{itemize}
  \item Using multiple stochastic integrals. In this case
        $I_{G}$ is constructed by representation of
        any function from $L^2({\mathcal S}^{'},\rho_{G})$
        as an infinite sum of pairwise
        orthogonal multiple sto\-cha\-stic integrals with respect
        to the
        Brownian motion $\{B_t\}_{t\in{\mathbb R}}$, see e.g.
        \cite{Hida75, I51, HKPS93}.
  \item Using Jacobi fields approach. Now $I_{G}$ is the Fourier transform
        of the free field, i.e., a certain family of commuting selfadjoint
        operators that act in the Fock space
        $F(L^2({\mathbb R}))$ and have the Jacobi
        structure, see for instance \cite{KS75, BK88}.
  \item Using the  system of infinite-dimensional Hermite polynomials which are
        orthogonal (in terms of the Fock space $F(L^2({\mathbb R}))$, see below)
        with respect to the Gaussian  measure $\rho_G$,
        see e.g. \cite{BK88, HKPS93}.
\end{itemize}
Our investigation is connected with the third way of the
construction of the Wiener-It\^{o}-Segal isomorphism $I_{G}$. Let us
look at this way more closely.

We consider the function
$$
   H(x,\varphi):=e^{\langle x,\varphi\rangle-
   \frac{1}{2}\|\varphi\|_{L_{{\mathbb C}}^2({\mathbb R})}^{2}},
   \qquad x\in{\mathcal S}{\,'},\quad \varphi\in{\mathcal S}_{{\mathbb C}}.
$$
It is  well known that $H$ is the {\it generating function} for
the {\it infinite-dimensional Hermite polynomials}
$H_n(x)\in({\mathcal S}^{\,'})^{\mathbin{\widehat{\otimes}} n}$
which are defined from the decomposition
$$
  H(x,\varphi)=\sum_{n=0}^{\infty}\frac{1}{n!}\langle\varphi^{\otimes
  n},H_n(x)\rangle,
$$
where symbol $\otimes$  denotes the tensor power.
The polynomials $H_n(x)$ are {\it orthogonal in the space} $L^2({\mathcal S}^{\,'},\rho_{G})$ in terms of the Fock
space $F(L^2({\mathbb R}))$,
\begin{equation}\label{eq.2}
   \int_{{\mathcal S}^{\,'}}\langle\varphi_n,H_n(x)\rangle
   \overline{\langle\psi_m,H_m(x)\rangle}\,
   d\rho_{G}(x)
   =\delta_{n,m}n!(\varphi_n,\psi_n)_{{\mathcal F}_n(L^2({\mathbb R}))},
\end{equation}
$$
   \varphi_n\in{\mathcal S}_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n},
   \quad \psi_m\in{\mathcal S}_{{\mathbb C}}^{\mathbin{\widehat{\otimes}}
   m},\qquad n,m\in{\mathbb Z}_+,
$$
and the mapping
\begin{equation*}
      F(L^2({\mathbb R}))\supset {\mathcal F}_{\mathop{\rm fin\,}}({\mathcal S})\ni
      \varphi=(\varphi_n)_{n=0}^{\infty}\mapsto
      (I_{G}\varphi)(\cdot):=\sum_{n=0}^{\infty}\langle\varphi_n,H_n(\cdot)\rangle\in
      L^2({\mathcal S}^{\,'},\rho_{G})
\end{equation*}
after being extended by continuity to the whole space
$F(L^2({\mathbb R}))$ is the Wiener-It\^{o}-Segal isomorphism. Here
${\mathcal F}_{\mathop{\rm fin\,}}({\mathcal S})$ denotes the set of
all finite sequences $(\varphi_n)_{n=0}^{\infty}$ such that each
$\varphi_n$ belongs to ${\mathcal S}_{{\mathbb
C}}^{\mathbin{\widehat{\otimes}} n}$.

With the help of the Wiener--It\^{o}--Segal isomorphism $I_{G}$,
spaces of test and generalized functions are constructed and
investigated. These spaces are obtained as the $I_{G}$-image of some
rigging of the Fock space $F(L^2({\mathbb R}))$:
$$
   \begin{array}{ccccc}
    {\mathcal F}_- & \supset & F(L^2({\mathbb R}))
    & \supset &  {\mathcal F}_+ \\
     &  &  &  &  \\
      &         &\downarrow I_{G}  &         &  \downarrow I_{G} \\
     &  &  &  & \\
         {\mathcal H}_-        & \supset & L^2({\mathcal S}^{\,'},\rho_{G}) & \supset
         &{\mathcal H}_+.
   \end{array}
$$
Here ${\mathcal F}_+$ is a certain Fock space densely and
continuously embedded into $F(L^2({\mathbb R}))$, ${\mathcal F}_-$
is the negative space with respect to the positive space
${\mathcal F}_+$ and the zero space $F(L^2({\mathbb R}))$. By
definition the space of test functions $
   {\mathcal H}_+:=I_{G}{\mathcal F}_+
$ is the $I_{G}$-image of the Fock space ${\mathcal F}_+$ with
topology inducted by the topology of ${\mathcal F}_+$, the space of
generalized functions ${\mathcal H}_-:=({\mathcal H}_+)^{'}$ is the
dual of ${\mathcal H}_+$ with respect to $L^2({\mathcal
S}^{\,'},\rho_{G})$. Note that we can extend the isomorphism
$I_{G}:F(L^2({\mathbb R}))\to L^2({\mathcal S}^{\,'},\rho_{G})$ to
the isomorphism between the negative Fock space ${\mathcal F}_-$ and
the space of generalized functions ${\mathcal H}_-$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Biorthogonal approach}

In this section we  give the basic idea of the biorthogonal
approach. In order to make it more simple, at first we will consider
the corresponding theory of generalized functions  for a model
one-dimensional case and than briefly for infinite-dimensional.

\subsection{One-dimensional case}

At first we consider one-dimensional analogue of the Gaussian white
noise analysis. Then we describe the biorthogonal approach to a
generalization of such analysis.

\subsubsection{Gaussian case}

Let $\rho_{G}$ be a {\it Gaussian measure} on on the Borel
$\sigma$-algebra ${\mathcal B}({\mathbb R})$, its Fourier transform
has the form
$$
   \int_{{\mathbb R}}e^{ix\lambda}d\rho_{G}(x)=e^{-\tfrac{1}{2}\lambda^2},
   \qquad \lambda\in{\mathbb R}.
$$

In this case we have the well-known {\it generating function}
$$
   H(x,\lambda):=e^{x\lambda-\tfrac{1}{2}\lambda^2}=
   \sum_{n=0}^{\infty}\frac{\lambda^n}{n!}H_n(x),\qquad
   x\in{\mathbb R},\quad\lambda\in{\mathbb C},
$$
for the  {\it Hermite polynomials} $H_n$ that are orthogonal with
respect to $\rho_{G}$. More precisely, we have, for all
$n,m\in{\mathbb Z}_+$,
$$
   \int_{{\mathbb R}}H_n(x)\overline{H_m(x)}\,d\rho_{G}(x)
   =\delta_{n,m}n!.
$$

Now the role of the Fock space $F(L^2({\mathbb R}))$ plays the
$l^2$-space
$$
   l^2:=\Big\{f=(f_n)_{n=0}^{\infty},\,f_n\in{\mathbb C}\,\Big|\,
   \|f\|_{l^2}^{2}:=\sum_{n=0}^{\infty}|f_n|^2n!<\infty\Big\}
$$
and an analogue of the Wiener-It\^{o}-Segal isomorphism has the form
$$
   l^2\ni f=(f_n)_{n=0}^{\infty}\mapsto
   (I_{G}f)(\cdot)=\sum_{n=0}^{\infty}f_nH_n(\cdot)\in
   L^2({\mathbb R},\rho_{G}).
$$

By analogy with the infinite-dimensional situation using the space
$l^2$ instead of the Fock space $F(L^2({\mathbb R}))$ and the
unitary map $I_{G}:l^2\to L^2({\mathbb R},\rho_{G})$ instead of the
Wiener-It\^{o}-Segal isomorphism we obtain spaces of test and
generalized functions of variables $x\in{\mathbb R}$ as the
$I_{G}$-image of a rigging of the space $l^2$.

Namely, for fixed $K>1$ and $q\in{\mathbb N}$ we denote
$$
   l^2_+(q):=\Big\{f=(f_n)_{n=0}^{\infty},\,f_n\in{\mathbb C}\,\Big|\,
   \|f\|_{l_+^2(q)}^{2}:=\sum_{n=0}^{\infty}|f_n|^2(n!)^2K^{qn}<\infty\Big\},
$$
$$
   l^2_+:=\mathop{\rm pr\,lim}_{q\in{\mathbb N}}l^2_+(q).
$$
Then the dual spaces of $l^2_+(q)$ and $l^2_+$ with respect to the
zero space $l^2$ are
$$
   l^2_-(q):=(l^2_+(q))^{'}=\Big\{f=(f_n)_{n=0}^{\infty},\,f_n\in{\mathbb C}\,\Big|\,
   \|f\|_{l_-^2(q)}^{2}:=\sum_{n=0}^{\infty}|f_n|^2K^{-qn}<\infty\Big\},
$$
$$
   l^2_-:=(l^2_+)^{'}=\mathop{\rm ind\,lim}_{q\in{\mathbb N}}l^2_-(q),
$$
respectively. Thus, for each $q\in{\mathbb N}$, we get a rigging
$$
   l^2_- \supset l^2_-(q) \supset  l^2   \supset l^2_+(q) \supset
   l^2_+.
$$

Using the unitary operator $I_{G}$, one defines spaces of test
functions
$$
   {\mathcal H}_+(q):=I_{G}l^2_+(q),\qquad
   {\mathcal H}_+:=I_{G}l^2_+=\mathop{\rm pr\,lim}_{q\in{\mathbb N}}{\mathcal H}_+(q),
$$
and their dual (with respect to the space $L^2({\mathbb
R},\rho_{G})$) spaces of generalized functions
$$
   {\mathcal H}_-(q):=({\mathcal H}_+(q))^{'},\qquad
   {\mathcal H}_-:=({\mathcal H}_+)^{'}
   =\mathop{\rm ind\,lim}_{q\in{\mathbb N}}{\mathcal H}_-(q).
$$
Hence, for each $q\in{\mathbb N}$, we have a rigging
$$
   {\mathcal H}_-  \supset  {\mathcal H}_-(q)  \supset
   L^2({\mathbb R},\rho_{G})  \supset  {\mathcal H}_+(q)  \supset {\mathcal H}_+
$$
with pairing between test and generalized functions provided by
integration with respect to the Gaussian measure $\rho_{G}$ on
${\mathbb R}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Biorthogonal case}

Let $\rho$ be a  Borel probability measure on ${\mathbb R}$,
$L^2({\mathbb R},\rho)$ be  the corresponding $L^2$-space. Our
purpose is to construct some class of test and generalized functions
on ${\mathbb R}$ with pairing provided by integration with respect
to $\rho$. We will try to construct these classes functions on
${\mathbb R}$ in a way parallel to the Gaussian case, but using a
certain biunitary mapping instead of the Wiener-It\^{o}-Segal
isomorphism.

Let us consider instead of $H(x,\lambda)$ a fixed function
$$
   {\mathbb R}\times{\mathbb C}\ni \{x,\lambda\}\mapsto
   h(x,\lambda)\in {\mathbb C}
$$
such that for each $\lambda$ from some neighborhood $B_0$
%$=\{\mu\in{\mathbb C}\,|\,|\mu|<r,r>0\}$
of zero in ${\mathbb C}$
the function ${\mathbb R}\ni x\mapsto h(x,\lambda)\in {\mathbb C}$
is continuous and for every $x\in {\mathbb R}$
$$
  h(x,\lambda)=\sum_{n=0}^{\infty}\frac{\lambda^n}{n!}h_n(x),
  \qquad \lambda\in B_{0}.
$$
We additionally assume that $h(\cdot,\lambda)$ is locally bounded
uniformly with respect to $\lambda$ on any closed ball inside of
$B_0$, and that $h(x,0)=1$ for all $x$ from ${\mathbb R}$. In our
consideration the role of the function $h(x,\lambda)$ will be same
as the role of the generating function
$H(x,\lambda)=e^{x\lambda-\frac{1}{2}\lambda^2}$ for the Hermite
polynomials in the Gaussian case.

We denote by $C({\mathbb R})$ the linear space of all complex-valued
locally bounded (i.e. bounded on every ball in ${\mathbb R}$)
continuous functions on ${\mathbb R}$. It follows from the
properties of $h$ that for every $n\in\mathbb{Z}_+$ the function
${\mathbb R}\ni x\mapsto h_n(x) \in {\mathbb C}$ belongs to the
space $C({\mathbb R})$ and the mapping
$$
   l^2_+(q)\ni f=(f_n)_{n=0}^{\infty}\mapsto
   (I^hf)(\cdot):=\sum_{n=0}^{\infty}f_nh_n(\cdot)\in
   C({\mathbb R})
$$
is well-defined for each $q\in {\mathbb N}$ and sufficiently large
$K>1$ (we recall that $K$ is used in the definition of the space
$l^2_+(q)$). In what follows, we fix such $K>1$.

From the~general results, one has (see e.g. \cite{BT03}).

\begin{thm}\label{t.1}
       Let the above mentioned function $h$ be such that
\begin{itemize}
  \item $\|h_n\|_{L^2({\mathbb R},\rho)}\leq C^nn!$ for some $C>0$ and all
        $n\in{\mathbb Z}_+$.
  \item The linear span of the functions $\{h_n\}_{n=0}^{\infty}$
        is dense in the space $L^2({\mathbb R},\rho)$.
  \item $\|I^hf\|_{L^2({\mathbb R},\rho)}=0$ if and only if $f=0$ in $l^2_+(q),
        \;q\in{\mathbb N}$.
\end{itemize}
Then the $I^h$-image
$$
   {\mathcal H}_+^h(q):=I^h(l^2_+(q))
   =\Big\{f\in C({\mathbb R})\,\Big|\,\exists (f_n)_{n=0}^{\infty}\in l^2_+(q),
   f(x)=\sum_{n=0}^{\infty}f_nh_n(x)\Big\}
$$
of the space $l^2_+(q),\,q\in{\mathbb N}$, is a Hilbert space of
continuous functions with topology inducted by the topology of
$l^2_+(q)$. Moreover ${\mathcal H}_+^h(q)$ densely and continuously
embedded in $L^2({\mathbb R},\rho)$ and  we can construct a rigging
$$
   {\mathcal H}_-^h\supset{\mathcal H}_-^h(q)\supset
   L^2({\mathbb R},\rho)  \supset {\mathcal H}_+^h(q)\supset
   {\mathcal H}_+^h,
$$
$$
   {\mathcal H}_+^h:=I^{h}l^2_+
   =\mathop{\rm pr\,lim}_{q\in{\mathbb N}}{\mathcal H}_+^h(q),\qquad
   {\mathcal H}_-^h:=({\mathcal H}_+^h)^{'}
   =\mathop{\rm ind\,lim}_{q\in{\mathbb N}}{\mathcal H}_-^h(q).
$$
\end{thm}

Let all requirements of Theorem \ref{t.1} be fulfilled. It follows
from \cite{B98} that for the unitary operator
$I^h:l^2_{+}(q)\to{\mathcal H}_{+}^h(q)$ there exists a unique
determined unitary operator $I_{-}^{h}:l^2_{-}(q)\to{\mathcal
H}_{-}^h(q)$ such that
%this equality takes place
$$
   (I_{-}^h\xi,I^h\varphi)_{L^2({\mathbb R},\rho)}=(\xi,\varphi)_{l^{2}},
   \qquad \xi\in l^{2}_{-}(q),\quad\varphi\in l^{2}_{+}(q).
$$
The pair $\{I_-^h,I^h\}$ is called a {\it biunitary map}. This
biunitary map transfers the rigging of the space $l^2$ to a
rigging of the space $L^2({\mathbb R},\rho)$:
$$
  \begin{array}{ccccc}
   l^2_-(q) & \supset &  l^2 & \supset & l^2_+(q) \\
     &  &  &  &  \\
     \downarrow I_{-}^h    &         &      &         & \downarrow I^h \\
     &  &  &  & \\
        {\mathcal H}_-^h(q)        & \supset &
        L^2({\mathbb R},\rho) & \supset & {\mathcal H}_+^h(q).
   \end{array}
$$

Thus, in biorthogonal case the spaces of test and generalized
functions are constructed in a way parallel to the Gaussian case (as
the image of the rigging of the space $l^2$), but using the
biunitary map $\{I_-^h,I^h\}$ instead of the Wiener-It\^{o}-Segal
isomorphism. This give us a possible to develop the biorthogonal
white noise analysis by analogy to the Gaussian analysis. In
particular, we can give an inner description of the spaces of test
and generalized functions, construct in general situation an
$S$-transform, Wick multiplication etc, see e.g. \cite{BT03} for
more details.

Note that there arises {\it a natural question}, --- under which
conditions on $h$ the biunitary map $\{I_-^h,I^h\}$ is the unitary
map, i.e.,  the system of functions $\{h_n\}_{n=0}^{\infty}$
constitutes an {\it orthogonal basis} in the space $L^2({\mathbb
R},\rho)$?

The answer is the following \cite{BT04}.

\begin{thm}\label{t.2}
      The system of the functions $\{h_n\}_{n=0}^{\infty}$ with the
      generating function $h$ constitutes an orthogonal
      basis in the space $L^2({\mathbb R},\rho)$ if and only if the
      following conditions  hold:
\begin{itemize}
  \item $\|h_n\|_{L^2({\mathbb R},\rho)}\leq C^nn!$ for some $C>0$ and all
        $n\in{\mathbb Z}_+$.
  \item The linear span of the functions $\{h_n\}_{n=0}^{\infty}$
        is dense in the space $L^2({\mathbb R},\rho)$.
  \item For each $\lambda,\mu$ from some neighborhood of zero in ${\mathbb C}$
        $$
           \int_{{\mathbb R}}h(x,\lambda)\overline{h(x,\mu)}d\rho(x)=e^{\lambda\bar{\mu}}.
        $$
\end{itemize}
\end{thm}

It is possible to prove  that if all conditions of
Theorem~\ref{t.2} take place then all conditions of
Theorem~\ref{t.1}  also will take place (see \cite{BT04}). In
other words the orthogonal situation is a particular case of the
biorthogonal situation.

Consider more special situation.

\begin{ex}
Let $\rho$ be a Borel probability measure on ${\mathbb R}$ such
that
$$
   \int_{{\mathbb R}}e^{\varepsilon|x|}d\rho(x)<\infty\quad
   \text{for\,\,some}\quad\varepsilon>0,
$$
$h(x,\lambda)$ be a generating function for the {\it Schefer
polynomials} $h_n(x)$  (in another terminology, the generalized
Appel polynomials), that is,
\begin{equation}\label{eq.Schefer}
   h(x,\lambda):=\gamma(\lambda)e^{\alpha(\lambda)x}
   =\sum_{n=0}^{\infty}\frac{\lambda^n}{n!}h_n(x),
  \qquad x\in{\mathbb R},\quad\lambda\in{\mathbb C},
\end{equation}
where $\gamma$ and $\alpha$ are fixed analytic functions in some
neighborhood of $0\in{\mathbb C}$ such that
$\alpha(0)=0,\,\alpha^{'}(0)=1$ and $\gamma(0)=1$.

In this case the estimate
$$
   \|h_n\|_{L^2({\mathbb R},\rho)}\leq C^nn!\quad
   {\text {\rm for some}}\quad C>0\quad {\text {\rm and all}}\quad n\in{\mathbb Z}_+
$$
is automatically satisfied and the linear span of the functions
$\{h_n\}_{n=0}^{\infty}$ is dense in the space $L^2({\mathbb
R},\rho)$, see e.g. \cite{KSS97, KSWY98}. Hence, if the mapping
$$
   l^2_+(q)\ni f=(f_n)_{n=0}^{\infty}\mapsto
   (I^hf)(\cdot):=\sum_{n=0}^{\infty}f_nh_n(\cdot)\in
   L^2({\mathbb R},\rho)
$$
is injective then all requirements of Theorem \ref{t.1} are
fulfilled and
  we can construct the
corresponding theory of generalized functions.

{\it The next question is}: Which of the Schefer polynomials are
orthogonal?

The answer was given by Meixner \cite{M34} in 1934 (see also
\cite{Lytv1, Rodionova} for more details). There exist exactly
five type of orthogonal Schefer polynomials: the Hermite,
Charlier, Laguerre, Meixner and Meixner--Pollaczek polynomials,
which are orthogonal with respect to the Gaussian, Poissonian,
Gamma, Pascal and Meixner measures respectively.
\end{ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Some useful tools in biorthogonal analysis}

Let $\rho$ be a Borel probability measure on ${\mathbb R}$ and
$$
   h(x,\lambda)=\sum_{n=0}^{\infty}\dfrac{\lambda^n}{n!}h_n(x),
  \qquad x\in{\mathbb R},\quad\lambda\in{\mathbb C},
$$ be a fixed function such that Theorem~\ref{t.1} takes place.

$\bullet$ \textit{Annihilation and creation operators}. The
annihilation operator $\partial$ acts continuously in the space of
test functions ${\mathcal H}_+^h$ by the formula
$$
   \partial h_n:=nh_{n-1},\qquad \partial h_0:=0.
$$
The creation operator $\partial^+$ is by definition the adjoint to
$\partial$ with respect to the zero space $L^2({\mathbb
R},\rho_{G})$ and acts continuously in the space of generalized
functions ${\mathcal H}_-^h$.

These operators play an essential role in our considerations.
Using them we investigate the spaces of test and generalized
functions, construct generalized translation operators, extended
stochastic integral (in infinite-dimensional case) etc (see e.g.
\cite{BT03, BT04, T04, T04a, ABT07} and references therein). Note
that in the Gaussian case the annihilation operator
$\partial$ % $\partial=\dfrac{d}{dx}$
is the derivative and $\partial+\partial^+$ as operator in the
space $L^2({\mathbb R},\rho_{G})$ is the operator of
multiplication by $x$.

$\bullet$ {\it $S$-transform}. For each $\xi\in {\mathcal H}_{-}^h$, the $S$-transform
is defined by the
formula
$$
   (S\xi)(\lambda):=\langle\!\langle\xi,h(\cdot,\overline{\lambda
   })\rangle\!\rangle,
$$
where $\lambda$ belongs to a neighborhood of zero in ${\mathbb C}$,
$\langle\!\langle\cdot\,,\cdot\rangle\!\rangle$ is the dual pairing
between elements of ${\mathcal H}_{-}^h$ and ${\mathcal H}_{+}^h$
generated by the scalar product in the space $L^2({\mathbb
R},\rho)$.

Each generalized function $\xi\in {\mathcal H}_{-}^h$ is uniquely determined by its
$S$-transform. More exactly, let $\mathop{\rm {Hol}}_{0}(\mathbb{C})$ denotes the
set of all (germs) of functions which are holomorphic in a
neighborhood of zero in $\mathbb{C}$. According to \cite{BK96}  the S-transform is a one-to-one map
between ${\mathcal H}_{-}^h$ and   $\mathop{\rm {Hol}}_{0}(\mathbb{C} )$.

$\bullet$ {\it Wick multiplication}.
Taking into account that $\mathop{\rm {Hol}}_{0}(\mathbb{C})$ is an algebra of
analytic functions with ordinary algebraic operations, we can
define a Wick product $\xi\lozenge\eta$ of $\xi,\eta\in {\mathcal H}_-^h$
through the formula
\begin{equation*}
   \xi\lozenge\eta:=S^{-1}(S\xi\cdot S\eta)
\end{equation*}
and make ${\mathcal H}_-^h$ an algebra with such
multiplication.

Using this multiplication we can construct the elements of Wick
calculus. In Gaussian white noise analysis such calculus has found
numerous applications, in particular, in fluid mechanics and
financial mathematics, see e.g. \cite{EH03, HOUZ96} for more
details.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Infinite-dimensional case}

Now we start with a fixed family $(H_{p})_{p\in{\mathbb Z}_+}$ of
{\it real} separable Hilbert spaces $H_{p}$ such that for all $p\in
{\mathbb Z}_+$ the space $H_{p+1}$ is densely embedded in $H_p$, and
this embedding is quasinuclear, i.e., the Hilbert-Schmidt type. So,
one can construct the rigging of the space $H_0$,
\begin{equation}\label{eq.2.9}
      \Phi^{\,'}:=\mathop{\rm ind\,lim}_{p\in {\mathbb Z}_+}H_{-p}
      \supset H_{-p}\supset H_{0}
      \supset H_{p}\supset
      \mathop{\rm pr\,lim}_{p\in {\mathbb Z}_+}H_{p}=:\Phi,
\end{equation}
where $H_{-p}$ is the dual space to $H_{p}$ with respect to the
zero space $H_{0}$. We denote by $\langle\cdot\, , \cdot\rangle$
the dual pairing between elements of $H_{-p}$ and $H_{p}$ inducted
by the scalar product in $H_{0}$. As earlier we will preserve
this notation  for tensor powers and complexifications of spaces.

For each $p\in{\mathbb Z}$ we introduce a weighted symmetric Fock
space ${\mathcal F}(H_p,\tau)$ over $H_p$ with a fixed weight
$\tau=(\tau_n)_{n=0}^{\infty},\;\tau_n>0$, by setting
\begin{equation*}\label{eq.2.11}
      \begin{split}
            {\mathcal F}(H_p,\tau):
           &
            =
            \bigoplus_{n=0}^{\infty}{\mathcal F}_n(H_p)\tau_n\\:
           &
            =\Big\{f=(f_n)_{n=0}^{\infty},\; f_n\in {\mathcal F}_n(H_p)\,\Big|\,
            \|f\|_{{\mathcal F}(H_p,\tau)}^2
            =\sum_{n=0}^{\infty}\|f_n\|_{{\mathcal F}_n(H_p)}^{2}
            \tau_n<\infty\Big\},
      \end{split}
\end{equation*}
where the $n$-particle Fock space
$$
   {\mathcal F}_n(H_p)
   :=H_{p,{\mathbb C}}^{\mathbin{\widehat{\otimes}}n}
   \qquad (H_{p,{\mathbb C}}^{\mathbin{\widehat{\otimes}} 0}:={\mathbb C})
$$
is equal to the $n$-th symmetric tensor power
$\mathbin{\widehat{\otimes}}$ of the complexification
$H_{p,{\mathbb C}}$ of the real space $H_{p}$. Using rigging
(\ref{eq.2.9}) and the weight
$$
      \tau(q)=((n!)^2K^{qn})_{n=0}^{\infty},
      \qquad q\in{\mathbb N},
$$
with fix $K>1$ we construct the rigging of the Fock space $
   F(H_0):={\mathcal F}(H_{0},(n!)_{n=0}^{\infty}),
$
\begin{equation*}
      {\mathcal F}(-p,-q)\supset F(H_0) \supset {\mathcal F}(p,q),
\end{equation*}
where
$$
   {\mathcal F}(-p,-q):={\mathcal F}(H_{-p},(K^{-qn})_{n=0}^{\infty}),
   \qquad
   {\mathcal F}(p,q):={\mathcal F}(H_{p},((n!)^2K^{qn})_{n=0}^{\infty})
$$
are dual with respect to the zero space $F(H_0)$.

Let $\rho$ be a fixed Borel probability measure on $\Phi^{\,'}$,
$L^2(\Phi^{\,'},\rho)$ be the corresponding space of square
integrable functions. Our goal is to construct some class of test
and generalized functions on $\Phi^{\,'}$ with pairing generated by
the scalar product in $L^2(\Phi^{\,'},\rho)$. We will try to
construct this class functions on $\Phi^{\,'}$ in a way parallel to
the Gaussian and above stated one-dimensional cases.

Let $B_0$ be a neighborhood  of zero in the space $\Phi_{{\mathbb
C}}$. Instead of the generating function for the
infinite-dimensional Hermite polynomials we will take a function
\begin{equation*}
      \Phi^{'}\times B_0\ni\{ x, \varphi \} \mapsto  h(x,\varphi)\in {\mathbb C}
\end{equation*}
such that for each $\varphi\in B_0$ the function $h(\cdot ,
\varphi)$ is continuous and for each  $x\in \Phi^{\,'}$ the
function $h(x,\cdot)$ is analytic in $B_0$, i.e., has the
representation
\begin{equation*}%\label{eq.3.1}
      h(x,\varphi)=\sum_{n=0}^{\infty }\frac{1}{n!}\langle
      \varphi^{\otimes n}, h_n(x) \rangle,\qquad
      h_n(x)\in (\Phi_{{\mathbb C}}^{\,'})^{\mathbin{\widehat{\otimes}} n},
\end{equation*}
for all $\varphi$ from $B_0$. We additionally assume that
$h(\cdot,\varphi)$ is locally bounded uniformly with respect to
$\varphi$ on any closed ball inside of $B_0$, and that $h(x,0)=1$
for all $x$ from $\Phi^{\,'}$ (see \cite{BT03}, Sections 2.3, for
more details).

Due to such properties of $h$ one can show that for each
$\varphi_n\in\Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n}$ the
functions
\begin{equation*}
   \Phi^{\,'}\ni x\mapsto \langle \varphi_n,h_n(x)\rangle
   \in{\mathbb C}
\end{equation*}
belong to the space $C(\Phi^{\,'})$ of all complex-valued locally
bounded continuous functions on $\Phi^{\,'}$. Moreover, there
exist $p,q\in{\mathbb N}$ and $K>1$ (we recall that $K$ is used in
the definition of space ${\mathcal F}(p,q)$) such that the mapping
\begin{equation*}
      {\mathcal F}(p,q)\ni f=(f_n)_{n=0}^{\infty}\mapsto
      (I^hf)(\cdot):=\sum_{n=0}^{\infty}
      \langle f_n,h_n(\cdot)\rangle\in C(\Phi^{\,'})
\end{equation*}
is well-defined. In what follows, we fix such $p,q\in{\mathbb N}$
and $K>1$.

According to \cite{BT03} we have.

\begin{thm}\label{t.3}
       Let the above mentioned function $h$ be such that
\begin{itemize}
  \item $\|\;\|h_n(\cdot)\|_{{\mathcal F}_n(H_{-p})} \|_{L^2(\Phi^{'},\rho)}
        \leq C^nn!$ for some $C>0$ and all
        $n\in{\mathbb Z}_+$.
  \item The linear span of the set of functions
        $$\{\langle \varphi_n,h_n(\cdot)\rangle\in L^2(\Phi^{\,'},\rho)\,
        |\, \varphi_n\in
        \Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n},\, n\in{\mathbb Z}_+\}$$
        is dense in the space $L^2(\Phi^{\,'},\rho)$.
  \item $\|I^hf\|_{L^2(\Phi^{\,'},\rho)}=0$ if and only if $f=0$ in ${\mathcal F}(p,q)$.
\end{itemize}
Then the $I^h$-image
\begin{equation*}
\begin{split}
  {\mathcal H}^h(p,q):
  &
  =I^h({\mathcal F}(p,q))\\
  &
  =\Big\{f\in C(\Phi^{\,'})\,\Big|\,\exists (f_n)_{n=0}^{\infty}
  \in{\mathcal F}(p,q),\,f(x)=\!\sum_{n=0}^{\infty}\langle f_n,
  h_n(x)\rangle\Big\}
\end{split}
\end{equation*}
of the space ${\mathcal F}(p,q)$ is a Hilbert space of continuous
functions with topology inducted by the topology of ${\mathcal
F}(p,q)$. Moreover ${\mathcal H}^h(p,q)$ densely and continuously
embedded in $L^2(\Phi^{\,'},\rho)$ and we can construct a rigging
$$
   {\mathcal H}^h(-p,-q)\supset L^2(\Phi^{\,'},\rho)  \supset {\mathcal H}^h(p,q)
$$
with pairing between  elements of ${\mathcal
H}^h(-p,-q):=({\mathcal H}^h(p,q))^{\,'}$ and ${\mathcal
H}^h(p,q)$ provided by integration with respect to the measure
$\rho$.
\end{thm}

Under the conditions of Theorem \ref{t.3}, for the unitary operator
$$I^h:{\mathcal F}(p,q)\to{\mathcal H}^{h}(p,q)$$ there exists a
unique determined unitary operator $$I_{-}^h:{\mathcal
F}(-p,-q)\to{\mathcal H}^h(-p,-q)$$ such that
$$
   (I_{-}^h\xi,I^h\varphi)_{L^2(\Phi^{\,'},\rho)}=(\xi,\varphi)_{F(H_0)},
   \qquad \xi\in {\mathcal F}(-p,-q),\quad\varphi\in {\mathcal F}(p,q).
$$
So, we have a {\it biunitary map} $\{I_-^h,I^h\}$. This map
transfers the rigging of the space $F(H_0)$ to a rigging of the
space $L^2(\Phi^{\,'},\rho)$:
$$
  \begin{array}{ccccc}
     {\mathcal F}(-p.-q) & \supset &  F(H_0) & \supset & {\mathcal F}(p,q) \\
     &  &  &  &  \\
     \downarrow I_{-}^h    &         &      &         & \downarrow I^h \\
     &  &  &  & \\
        {\mathcal H}^h(-p,-q)        & \supset &
        L^2(\Phi^{\,'},\rho) & \supset & {\mathcal H}^h(p,q).
   \end{array}
$$

Thus, in biorthogonal case the spaces of test and generalized
functions are constructed in a way parallel to the Gaussian case (as
the image of the rigging of the Fock space $F(H_0)$), but using the
biunitary map $\{I_-^h,I^h\}$ instead of the Wiener-It\^{o}-Segal
isomorphism $I_{G}$.

The infinite-dimensional analogue of Theorem \ref{t.2} holds,
see \cite{BT03, BT04}.

\begin{thm}\label{t.4}
      The mapping
\begin{equation}\label{eq.3.4}
      F(H_0)\ni
      f=(f_n)_{n=0}^{\infty}\mapsto
      (I^hf)(\cdot):=\sum_{n=0}^{\infty}\langle f_n,h_n(\cdot)\rangle\in
      L^2(\Phi^{\,'},\rho)
\end{equation}
      is well-defined and unitary if and only if the
      following three conditions  hold:
\begin{itemize}
  \item $\|\;\|h_n(\cdot)\|_{{\mathcal F}_n(H_{-p})} \|_{L^2(\Phi^{'},\rho)}
        \leq C^nn!$ for some $C>0$ and all
        $n\in{\mathbb Z}_+$.
  \item The linear span of the set of functions
        $$\{\langle \varphi_n,h_n(\cdot)\rangle\in L^2(\Phi^{\,'},\rho)\,
        |\, \varphi_n\in \Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n},\,
        n\in{\mathbb Z}_+\}$$
        is dense in the space $L^2(\Phi^{\,'},\rho)$.
  \item For each $\varphi,\psi$
        from some neighborhood of zero in $\Phi_{{\mathbb C}}$
        \begin{equation*}\label{eq.2.81}
              \int_{\Phi^{\,'}} h(x,\varphi)\overline{h(x,\psi)}
              d\rho(x)=e^{(\varphi,\psi)_{H_{0,{\mathbb C}}}}.
        \end{equation*}
\end{itemize}
\end{thm}

Note that under the assumptions of Theorem \ref{t.4}
the functions
\begin{equation*}
   \Phi^{\,'}\ni x\mapsto \langle \varphi_n,h_n(x)\rangle
   \in{\mathbb C},\qquad
   \Phi^{\,'}\ni x\mapsto \langle \psi_m,h_m(x)\rangle
   \in{\mathbb C},
\end{equation*}
$$
   \varphi_n\in\Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n},
   \quad \psi_m\in\Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}}
   m},\qquad n,m\in{\mathbb Z}_+,
$$
are orthogonal in the space $L^2(\Phi^{\,'},\rho)$ in terms of the Fock space $F(H_0)$,
\begin{equation*}
      \int_{\Phi^{\,'}}\langle \varphi_n,h_n(x)\rangle
      \overline{\langle \psi_m,h_m(x)\rangle}d\rho(x)
      =\delta_{n,m}n!(\varphi_n,\psi_n)_{{\mathcal F}_n(H_0)},
\end{equation*}
and all requirements of Theorem~\ref{t.3} are fulfilled.

Let us consider the special case when $\Phi=\mathcal{S}$,
$H_0=L^2(\mathbb{R})$ and as a consequence
$\Phi^{'}=\mathcal{S}^{'}$. Let the function $h$ satisfies all
conditions of Theorem \ref{t.4} and
\begin{multline}\label{eq.6.11}
      \qquad\frac{\partial^n h(x,z_1\varphi_1+\cdots+z_n\varphi_n)}
      {\partial z_1\ldots\partial z_n}\bigg|_{z_1=\ldots=z_n=0\in\mathbb{C}}
      \\
      =\frac{\partial}{\partial
      z_1}h(x,z_1\varphi_1)\bigg|_{z_1=0\in\mathbb{C}}\ldots\;
      \frac{\partial}{\partial z_n}h(x,z_n\varphi_n)\bigg|_{z_n=0\in\mathbb{C}}
\end{multline}
for all $x\in \mathcal{S}^{'}$ and all
$\varphi_1,\ldots,\varphi_n\in\mathcal{S}$ such that $\hbox{\rm
{supp\,}}\varphi_i\cap\hbox{\rm {supp\,}}\varphi_j=\varnothing$ if
$j\neq i,\,$ $i,j\in\{1,\ldots,n\},\,n\in\mathbb{N}$. Then according
to \cite{ABT07} the mapping (\ref{eq.3.4}) is completely
characterized by the following properties:
\begin{enumerate}
  \item $I^h:F(L^2({\mathbb R}))\to L^2({\mathcal S}^{\,'},\rho)$
        is the unitary operator.
  \item $I^h(f_0,0,0,\ldots)=f_0$ for all $f_0\in{\mathbb C}$.
  \item For each $n\in{\mathbb N}$ and any disjoint sets
        $\alpha_1,\ldots, \alpha_n\in\mathcal{B}(\mathbb{R})$ of finite Lebesgue measure,
      \begin{equation*}
            \big(I^h(\underbrace{0,\ldots,0}_{n},\varkappa_{\alpha_1}\mathbin{\widehat{\otimes}}
            \cdots\mathbin{\widehat{\otimes}}\varkappa_{\alpha_n},0,0,\ldots)\big)(\cdot)
            =\langle h_1(\cdot),\varkappa_{\alpha_1}\rangle\ldots
            \langle h_1(\cdot),\varkappa_{\alpha_n}\rangle.
      \end{equation*}
\end{enumerate}
Note that in the case of the Gaussian measure $\rho:=\rho_{G}$ on
$\mathcal{B}({\mathcal S}^{\,'})$ the function
$$
   h(x,\varphi):=H(x,\varphi)=e^{\langle x,\varphi\rangle-
   \frac{1}{2}\|\varphi\|_{L_{{\mathbb C}}^2({\mathbb R})}^{2}},
   \qquad x\in{\mathcal S}{\,'},\quad \varphi\in{\mathcal S}_{{\mathbb C}},
$$
satisfies all conditions of Theorem \ref{t.4} and equality
(\ref{eq.6.11}).

Now we have an analog of the Example from Subsection 3.1.

\begin{ex}
Let $\rho$ be a Borel probability measure on $\Phi^{\,'}$ such
that
$$
   \int_{\Phi^{\,'}}e^{\varepsilon\|x\|_{H_{-p}}}d\rho(x)<\infty\quad
   \text{for\,\,some}\quad\varepsilon>0\quad\text{and}\quad
   p\in{\mathbb N}.
$$
Schefer polynomials (\ref{eq.Schefer}) have the corresponding
infinite-dimensional counterpart and are defined by the Taylor
expansion of the generating function
$$
  h(x,\varphi):=\gamma(\varphi)e^{\langle\alpha(\varphi),x\rangle}
  =\sum_{n=0}^{\infty}\frac{1}{n!}\langle\varphi^{\otimes
  n},h_n(x)\rangle,
$$
where $\gamma$ and $\alpha$ are fixed analytic functions in some
neighborhood of zero in $\Phi_{{\mathbb C}}$ such that
$\alpha(0)=0,\,\gamma(0)=1$ and $\alpha$ is invertible in a
neighborhood of zero.

In this case the estimate
$$
   \|\;\|h_n(\cdot)\|_{{\mathcal F}_n(H_{-p})} \|_{L^2(\Phi^{'},\rho)}
        \leq C^nn!\quad
   {\text {\rm for some}}\quad C>0\quad {\text {\rm and all}}\quad n\in{\mathbb Z}_+
$$
is automatically satisfied and the linear span of the functions
$$
   \{\langle \varphi_n,h_n(\cdot)\rangle\in L^2(\Phi^{\,'},\rho)\,
   |\, \varphi_n\in \Phi_{{\mathbb C}}^{\mathbin{\widehat{\otimes}} n},\,
   n\in{\mathbb Z}_+\}
$$
is dense in the space $L^2({\mathbb R},\rho)$, see e.g. \cite{KSS97,
KSWY98}. Hence, if the mapping
\begin{equation*}
      {\mathcal F}(p,q)\ni f=(f_n)_{n=0}^{\infty}\mapsto
      (I^hf)(\cdot):=\sum_{n=0}^{\infty}
      \langle f_n,h_n(\cdot)\rangle\in L^2(\Phi^{'},\rho)
\end{equation*}
is injective then all requirements of Theorem \ref{t.3} are
fulfilled and we can construct the corresponding theory of
generalized functions (see \cite{AKS93, Us95, ADK96, LU96, KSS97,
KSSU98, KL00, B02, KKO02, Rodionova, Lytv06} for more detailed
acount).

For the infinite-dimensional counterpart of the Hermite, Charlier,
Laguerre, Me\-ix\-ner and Meixner--Pollaczek polynomials the
orthogonality preserves in the following sense:
\begin{itemize}
  \item  In the Gaussian and Poisson cases the orthogonality of
  the Hermite and Charlier polynomials respectively is given in
  terms of the Fock space (relation type (\ref{eq.2})), see e.g.
  the books \cite{Hida75, BK88, HKPS93}  and articles
  \cite{IK88, Us95, KSSU98, B00a, KKO02, BT04}.
  Note that the study of Poisson white noise analysis was initiated
  by Y.~Ito and I.~Kubo \cite{IK88} in 1988. They were the first to
  construct spaces of test and generalized functions of Poisson
  white noise, to study them and some operators acting in these
  spaces.
  \item  In the Gamma, Pascal and Meixner cases the orthogonality of
  the corresponding polynomials is more complicated and is given
  in terms of the so-called ``extended Fock space'',
  see for instance \cite{KSSU98, KL00, BM01, B02, BLM03,
  BM03, Lytv1, Lytv2, Rodionova, Lytv06}.
\end{itemize}
\end{ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}

\bibitem{ABT07}
 {S. Albeverio, Yu. M. Berezansky, V. Tesko},
 \textit{A generalization of an extended stochastic integral.}
 {Ukrainian Math. J.}
 \textbf{59}
 {(2007)},
 {no.~5},
 {588--617}.
% [ABT07]

\bibitem{ADK96}
 {S. Albeverio, Yu. L. Daletsky, Yu. G. Kondratiev, and L. Streit},
 \textit{Non-Gaussian infinite-dimensional analysis.}
 {J. Funct. Anal.}
 \textbf{138}
 {(1996)},
 {no.~2},
 {311--350}.
% [ADK96]

\bibitem{AKS93}
 {S. Albeverio, Yu. G. Kondratiev,  and  L. Streit},   %authors
 \textit{How to generalize white noise analysis to non-Gaussian measures.}%paper
 {Proc. Symp. Dynamics of Complex and Irregular Systems
 (Bielefeld, 1991);
 Bielefeld Encount. Math. Phys., VIII},   %book
 { World Sci. Publ., River Edge, NJ}, %bookinfo
% {vol.~120}, %vol
 {1993},     %year
 {pp.~ 120--130}. %pages
% [AKS93]

\bibitem{B91}
 {Yu. M. Berezansky},   %authors
 \textit{Spectral approach to white noise analysis.}%paper
 {Proc. Symp. Dynamics of Complex and Irregular Systems
  (Bielefeld, 1991); Bielefeld Encount. Math. Phys., VIII},   %book
 {World Sci. Publ., River Edge, NJ}, %bookinfo
% {vol.~120}, %vol
 {1993},     %year
 {pp.~131--140}. %pages
% [B91]

\bibitem{B98}
 {Yu.~M. Berezansky},
 \textit{ Infinite dimensional non-Gaussian analysis connected
 with generalized translation operators.}
 {Analysis on infinite-dimensional Lie groups and algebras (Marseille, 1997)},
 {World Sci. Publ., River Edge, NJ},
 {1998},
 {pp.~22--46}.
% [B98]

\bibitem{B98b}
 {Yu.~M. Berezansky},
 \textit{Commutative Jacobi fields in Fock space.}
 {Integr. Equ. Oper. Theory.}
 \textbf{30}
 {(1998)},
 {no.~2},
 {163--190}.
% [B98b]

\bibitem{B98e}
 {Yu.~M. Berezansky},
 \textit{Direct and inverse spectral problems for a Jacobi field.}
 {St. Petersburg Math. J.}
 \textbf{9}
 {(1998)},
 {no.~6},
 {1053--1071}.
% [B98e]

\bibitem{B98f}
 {Yu.~M. Berezansky},
 \textit{On the theory of commutative Jacobi fields.}
 {Methods Funct. Anal.  Topopology}
 \textbf{4}
 {(1998)},
 {no.~1},
 {1--31}.
% [B98f]

\bibitem{B00}
 {Yu.~M. Berezansky},
 \textit{Spectral theory of commutative Jacobi fields: direct and inverse problems.}
 {Fields Inst. Conmun.}
 \textbf{25}
 {(2000)},
 {211--224}.
% [B00]

\bibitem{B00a}
 {Yu.~M. Berezansky},
 \textit{Poisson measure as the spectral measure of Jacobi field.}
 { Infin. Dimens. Anal. Quantum Probab. Relat. Top.}
 \textbf{3}
 {(2000)},
 {no.~1},
 {121--139}.
% [B00a]

\bibitem{B02}
 {Yu. M. Berezansky},
 \textit{Pascal measure on generalized
 functions and the corresponding generalized Meixner
 polynomials.}
 {Methods Funct. Anal.  Topopology}
 \textbf{8}
 {(2002)},
 {no.~1},
 {1--13}.
% [B02]

\bibitem{BK88}
 {Yu.~M.~Berezansky and Yu.~G.~Kondratiev},   %authors
 \textit{Spectral Methods in Infinite Dimensional Analysis.}  %book
 { Vols.~1, 2, Kluwer Academic Publishers}, %bookinfo
% {TBiMC},    %publ
 {Dordrecht--Boston--London},     %publaddr
 {1995}.     %year
 (Russian edition: Naukova Dumka, Kiev, 1988).
% [BK88]

\bibitem{BK95}
 {Yu.~M. Berezansky and Yu.~G. Kondratiev},
 \textit{Non-Gaussian analysis and hypergroups.}
 {Funct. Anal. Appl.}
 \textbf{29}
 {(1995)},
 {no.~3},
 {188--191}.
% [BK95]

\bibitem{BK96}
 {Yu.~M. Berezansky and Yu.~G. Kondratiev},
 \textit{Biorthogonal systems in hypergroups:
 an extension of non-Gaussian analysis.}
 {Methods Funct. Anal.  Topopology}
 \textbf{2}
 {(1996)},
 {no.~2},
 {1--50}.
% [BK96]

\bibitem{BLLy95}
 {Yu.~M. Berezansky, V.~O. Livinsky, and E. V. Lytvynov},
 \textit{A generalization of Gaussian white noise analysis.}
 {Methods Funct. Anal.  Topopology}
 \textbf{1}
 {(1995)},
 {no.~1},
 {28--55}.
% [BLLy95]

\bibitem{BLM03}
 {Yu. M. Berezansky, E. Lytvynov, and D. A. Mierzejewski},
 \textit{The Jacobi field of a Levy process.}
 {Ukrainian Math. J.}
 \textbf{55}
 {(2003)},
 {no.~5},
 {853--858}.
% [BLM03]

\bibitem{BM01}
 {Yu.~M. Berezansky and D. A. Mierzejewski},
 \textit{The chaotic decomposition for the Gamma field.}
 { Funct. Anal. Appl.}
 \textbf{35}
 {(2001)},
 {no.~4},
 {305--308}.
% [BM01]

\bibitem{BM03}
 {Yu.~M. Berezansky and D. A. Mierzejewski},
 \textit{ The Construction of the Chaotic Representation for the Gamma Field.}
 { Infin. Dimens. Anal. Quantum Probab. Relat. Top.}
 {\bf 6}
 {(2003)},
 {no.~1},
 {33--56}.
% [BM03]

\bibitem{BS73}
 {Yu. M. Berezansky and Yu. S. Samoilenko},   %authors
 \textit{ Nuclear spaces of functions of an infinite number of variables.} %paper
 {Ukrainian Math. J.} %journal
 \textbf{25} %vol
 {(1973)},   %year
 {no.~6},    %issue
 {723--737} %pages
 (in Russian).
% [BS73]

\bibitem{BT03}
 {Yu. M. Berezansky and V. A. Tesko},
 \textit{ Spaces of fundamental and generalized functions associated
 with generalized translation.}
 {Ukrainian Math. J.}
 \textbf{55}
 {(2003)},
 {no.~12},
 {1907--1979}.
% [BT03]

\bibitem{BT04}
 {Yu. M. Berezansky and V. A. Tesko},
 \textit{An orthogonal approach to the construction of the theory of
 generalized functions of an
 infinite number of variables, and Poissonian white noise analysis.}
 {Ukrainian Math. J.}
 \textbf{56}
 {(2004)},
 {no.~12},
 {1885--1914}.
%[BT04]

\bibitem{D91}
 {Yu. L. Daletsky},
 \textit{A biorthogonal analogy of
        the Hermite polynomials and the inversion of the Fourier transform
        with respect to a non-Gaussian measure.}
 {Funct. Anal. Appl.}
 \textbf{25}
 {(1991)},
 {no.~2},
 {138--40}.
% [D91]

\bibitem{EH03}
 {R. J. Elliott and J. van der Hoek},
 \textit{A general fractional white
         noise theory and applications to finance.}
 {Math. Finance}
 \textbf{13}
 {(2003)},
 {no.~2},
 {301--330}.
% [EH03]

\bibitem{Hida75}
 {T. Hida},   %authors
 \textit{Analysis of Brownian functionals.} %paper
 {Carleton Math. Lect. Notes}, %journal
 \textbf{13} %vol
 {(1975)}.   %year
% {no.~4},    %issue
% {101--107}. %pages
% [Hida75]

\bibitem{HKPS93}
 {T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit},
 \textit{White Noise. An Infinite Dimensional Calculus.}
 {Kluwer Academic Publishers},
 {Dordrecht},
 {1993}.
% [HKPS93]

\bibitem{HOUZ96}
 {H. Holden, B. \O ksendal, J. Ub\o e, and T. Zhang},   %authors
 \textit{Stochastic Partial Differential
         Equations. A Modeling, White Noise Functional Approach.}  %book
% {bookinfo}, %bookinfo
 {Birkheauser},    %publ
 {Boston},     %publaddr
 {1996}.     %year
% [HOUZ96]

\bibitem{I51}
 {K.Ito},
 \textit{Multiple Wiener Integral.}
 {J. Math. Soc. Japan}
 \textbf{3}
 {(1951)},
 {157--169}.
% [I51]

\bibitem{IK88}
 {Y.Ito and I. Kubo},
 \textit{Calculus on Gaussian and Poissonian white noise.}
 {Nagoya Math. J.}
 \textbf{111}
 {(1988)},
 {41--84}.
% [IK88]

\bibitem{Ka96}
 {N. A. Kachanovsky},
 \textit{Biortogonal Appell-like systems in a Hilbert space.}
 {Methods Funct. Anal.  Topopology}
 \textbf{2}
 {(1996)},
 {no.~3-4},
 {36--52}.
% [Ka96]

\bibitem{KaK99}
 {N. A. Kachanovsky and S. V. Koshkin},
 \textit{Minimality of Appell-like systems and embeddings of test function spaces
 in a generalization of white noise analysis.}
 {Methods Funct. Anal.  Topopology}
 \textbf{5}
 {(1999)},
 {no.~3},
 {13--25}.
% [KaK99]

\bibitem{KKO02}
 {Yu. G. Kondratiev, T. Kuna, M. J. Oliveira},
 \textit{Analytic aspects of Poissonian white noise analysis.}
 {Methods Funct. Anal.  Topopology}
 \textbf{8}
 {(2002)},
 {no.~4},
 {15--48}.
% [KKO02]

\bibitem{KL00}
 {Yu. G. Kondratiev and E. W. Lytvynov},
 \textit{Operators of Gamma white noise calculus.}
 {Infin. Dimens. Anal. Quantum Probab. Relat. Top.}
 \textbf{3}
 {(2000)},
 {no.~3},
 {303--335}.
% [KL00]

\bibitem{KSS97}
 {Yu. G. Kondratiev, J. L. Da Silva,  and L. Streit},
 \textit{Generalized Appell systems.}
 {Methods Funct. Anal.  Topopology}
 \textbf{3}
 {(1997)},
 {no.~3},
 {28--61}.
% [KSS97]

\bibitem{KSSU98}
 {Yu. G. Kondratiev, J. L. Da Silva, L. Streit, and G. F. Us},
 \textit{Analysis on Poisson and Gamma spaces.}
 { Infin. Dimens. Anal. Quantum Probab. Relat. Top.}
 \textbf{1}
 {(1998)},
 {no.~1},
 {91--117}.
% [KSSU98]

\bibitem{KSWY98}
 {Yu. G. Kondratiev, L. Streit, W. Westerkamp and J. Yan},
 \textit{Generalized functions in infinite-dimensional analysis.}
 { Hiroshima Math. J.}
 \textbf{28}
 {(1998)},
 {no.~2},
 {213--260}.
% [KSWY98]

\bibitem{KS75}
 {V. D. Koshmanenko and Ju. S. Samoilenko},
 \textit{The isomorphism between Fok space and a space of
 functions of infinitely many variables.}
 {Ukrainian Math. J.}
 \textbf{27}
 {(1975)},
 {no.~5},
 {669--674}
 (in Russian).
% [KS75]

\bibitem{Krein1}
 {M. G. Krein},
 {\textit Fundamental aspects of the representation theory of Hermitian
 operators with deficiency index $(m,m)$.}
 {Ukrainian Math. J.}
 \textbf{1}
 {(1949)},
 {no.~2},
 {3--66}
 (in Russian).

\bibitem{Krein2}
 {M. G. Krein},
 {\textit Infinite $J$-matrices and a matrix moment problem.}
 {Dokl. Acad. Nauk SSSR}
 \textbf{69}
 {(1949)},
 {no.~2},
 {125--128}
 (in Russian).

\bibitem{Kuo96}
 {H.-H. Kuo},   %authors
 \textit{White noise distribution theory.}  %book
% {bookinfo}, %bookinfo
 {CRC Press},    %publ
% {Kyiv},     %publaddr
 {1996}.     %year
% [Kuo96]

\bibitem{Kuo02}
 {H.-H. Kuo},   %authors
 \textit{White noise theory.} %paper
 {Handbook of Stochastic Analysis and
  Applications D. Kannan and V. Lakshmikantham (eds.)},   %book
% {bookinfo}, %bookinfo
%{vol.~120}, %vol
 {2002},     %year
 {pp.~107--158}. %pages
% [Kuo02]

\bibitem{Kuo02a}
 {H.-H. Kuo},   %authors
 \textit{A quarter century of white noise theory.} %paper
 {Quantum Information IV, T. Hida and K. Saito (eds.)},   %book
% {bookinfo}, %bookinfo
%{vol.~120}, %vol
 {2002},     %year
 {pp.~1--37}. %pages
% [Kuo02a]

\bibitem{L95}
 {E. W. Lytvynov},
 \textit{ Multiple Wiener integrals and non-Gaussian
 white noises: a Jacobi field approach.}
 {Methods Funct. Anal.  Topopology}
 \textbf{1}
 {(1995)},
 {no.~1},
 {61--85}.
% [L95]

\bibitem{Lytv1}
 {E. Lytvynov},
 \textit{Polynomials of Meixner's type in infinite dimensions --- Jacobi
 fields and orthogonality measures.}
 {J. Funct. Anal.}
 \textbf{200}
 {(2003)},
 {118--149}.
% [Lytv1]

\bibitem{Lytv2}
 {E. Lytvynov},
 \textit{Orthogonal decompositions for Levy processes with an applications
 to the Gamma, Pascal, and Meixner processes.}
 { Infin. Dimens. Anal. Quantum Probab. Relat. Top.}
 \textbf{6}
 {(2003)},
 {no.~1},
 {73--102}.
% [Lytv2]

\bibitem{Lytv06}
 {E. Lytvynov},   %authors
 \textit{Functional spaces and operators connected
 with some Levy noises.} %paper
 arXiv:math.PR/0608380 v1 15 Aug 2006.
% {Theory of Probability} %journal
% \textbf{70} %vol
% {(2004)},   %year
% {no.~4},    %issue
% {101--107}. %pages
%% [Lytv06]

\bibitem{LU96}
 {E. W. Lytvynov and G. F. Us},
 \textit{Dual Appell systems in non-Gaussian white noise calculus.}
 {Methods Funct. Anal.  Topopology}
 \textbf{2}
 {(1996)},
 {no.~2},
 {70--85}.
% [LU96]

\bibitem{M34}
 {J. Meixner},
 \textit{Orthogonale Polynomsysteme mit einem besonderen
         Gestalt der erzeugenden Funktion.}
 {J. London Math. Soc}
 \textbf{9}
 {(1934)},
 {no.~1},
 {6--13}.
% [M34]

%\bibitem{Mi03}
% {D. A. Mierzejewski},
% \textit{Generalized Jacobi Fields.}
% {Methods Funct. Anal.  Topopology}
% \textbf{9}
% {(2003)},
% {no.~1},
% {80--100}.
%% [Mi03]

\bibitem{Rodionova}
 {I. Rodionova},
 \textit{Analysis connected with generating functions of
 exponential type in one and infinite dimensions.}
 {Methods Funct. Anal.  Topopology}
 \textbf{11}
 {(2005)},
 {no.~3},
 {275--297}.
% [Rodionova]

\bibitem{T04}
 {V. A. Tesko},
 \textit{A construction of generalized translation operators.}
 {Methods Funct. Anal.  Topopology}
 \textbf{10}
 {(2004)},
 {no.~4},
 {86--92}.
% [T04]

\bibitem{T04a}
 {V. A. Tesko},   %authors
 \textit{On spaces that arise in the construction of
 infinite-dimensional analysis according to a biorthogonal scheme.} %paper
 {Ukrainian Math. J.} %journal
 \textbf{56} %vol
 {(2004)},   %year
 {no.~7},    %issue
 {1166--1181}. %pages
% [T04a]

\bibitem{Us95}
 {G. F. Us},
 \textit{Dual Appel systems in Poissonian analysis.}
 {Methods Funct. Anal.  Topopology}
 \textbf{1}
 {(1995)},
 {no.~1},
 {93--108}.
% [Us95]

\end{thebibliography}

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%%%\subsection*{Acknowledgment}
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\end{document}
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