\section{Introduction} We study causal dynamic approximation of
non-bandlimited  discrete  time  processes (i.e., time series) by band-limited discrete time
 processes. This task has many practical applications and was
studied intensively, especially for continuous time processes.
Continuous time band-limited processes play a special role in  signal processing; by the Nyquist-Shannon-Kotelnikov interpolation theorem,
these signals can be completely recovered from their values at the sampling points. These processes are analytic; they allow a unique extrapolation from any interval, and they  are uniquely defined by their past. In this sense, these process are predicable.

 For many applications, it is preferable to replace a process by its band-limited approximation.
In theory, a process can be converted to a band-limited process with a low-pass filter.     However,
 a ideal low-pass filter is non-causal; therefore, it cannot be applied for the process that is observable
 dynamically such that its future values are unavailable.
 Moreover, in continuous time setting, it was shown in \cite{rema}
 that the distance of the set of  ideal low-pass
filters from the set of all causal filters is positive; see also discussion in \cite{D12c}.
Respectively,  causal smoothing cannot convert a process into a band-limited one.
 There are many works devoted to causal smoothing and sampling, oriented on estimation and minimization of errors in $L_2$-norms or similar norms, especially in stochastic setting; see, e.g., Wiener \cite{W},
Szeg\"o \cite{Sz,Sz1}, Verblunsky \cite{V}, Kolmogorov \cite{K}, and more recent papers
\cite{jerry,pollock,PFG,PFG1,AU,S,Bin}.
  \index{\cite{jerry}, \cite{pollock}, \cite{PFG}, \cite{PFG1}, \cite{AU}.}
\par
   The present paper readdresses  the problem of causal  band-limited smoothing
   approximation.
 We consider approximation in the deterministic setting, i.e., pathwise, using
pathwise optimality criterion rather than criterion calculated via expectation.
This setting is attractive since it does not rely on the statistical properties of the underlying process. Further,
we consider an approximation that does not target the match of the values at any set of selected points; the error is not expected to be small.
This is different from the more common setting where the goal is to match
the approximating curve with underlying process
at certain sampling points (see, e.g., \cite{jerry,pollock,F94,FKR,LeeF}).
Our setting is closer to the setting from  \cite{PFG,PFG1,TH}.
In \cite{PFG,PFG1}, the point-wise matching error
was estimated for a sampling series and for a band-limited process representing
smoothed
underlying  continuous time process; the  estimate featured a
given error that vanishes. In \cite{TH}, the problem  of minimization of the total energy  of the approximating bandlimited process was considered; this causal approximation was constructed
 within  a given distance from  the original process smoothed by an ideal low-pass filter.
We consider approximation of the underlying process by a band-limited process directly using
different  methods, and we seek an approximation minimizing the $L_2$--norm of the error
 with respect to the original non-smoothed process. Another related result was obtained in \cite{F94}, where interpolation problem for absent sampling points  was considered in a setting with vanishing error, for a finite number of sampling points. In our setting, the main focus is on the minimization of a non-vanishing error and on
 the case of  infinite number of sampling points.
For the solution, we use  non-singularity  of special sinc matrices  obtained recently in \cite{LeeF}
for the solution of the so-called superoscillations problem for continuous time processes; see the references in  \cite{FKR,LeeF}. It can be noted that the setting in \cite{FKR,LeeF}
considers exact matching of the band-limited process and the underlying process in certain points, which is  different from our setting.

We consider the case of discrete time underlying  processes and discrete  time approximating discrete time  band-limited processes;
the values between fixed discrete times are  are not included into consideration. The discrete time  band-limited processes are  still predictable in some weak sense; see, e.g.,  \cite{D10,D12a,D12b}. However, this setting imposes certain restrictions. In particular, it does not allow to consider continuously
variable locations of the sampling points, as is common in sampling analysis of continuous time processes; see, for example, \cite{PFG,PFG1,BS,FKR,LeeF,M}. For discrete time processes or sequences,  it is not obvious how to
define an analog of the continuous time analyticity.
Further, the predicting horizon for underlying continuous  time processes can be selected to be arbitrarily  small, such as in the model considered  in \cite{BS}. This possibility is absent for discrete time processes.
\par
We consider mean-square optimal approximation of  the known historical path of the underlying process by the traces of the band-limited discrete time processes.
We obtain sufficient conditions of existence and uniqueness of an
optimal approximating band-limited process; the optimal process is derived in
time domain in a form of sinc series.  An unique extrapolation to future times of this process  can be interpreted as an optimal
forecast. This process can
be interpreted as the output of a  causal and linear filter.
To accommodate the current
flow of observations, the coefficients of the sinc series have to
be changed dynamically; therefore,  the corresponding filter is not time
invariant.   It can be noted that, for some problems, time
invariance for a filter is not crucial. For example, a typical
approach to forecasting in finance is to approximate the known path
of the stock price process by a process that has a unique
extrapolation that can be used as a forecast. This
procedure can be done at current time; it is not required that
the same forecasting rule will be applied at future times.
\index{\par
The results of this paper  were  presented
by the author on  the 2nd International Conference on
Operations Research and Enterprise Systems (ICORES 2013).}
\section{Definitions}
We use notation ${\rm sinc\,}(x)=\sin(x)/x$.
\par
We denote by ${\mathbb{Z}}$  the set of all integers. \par
We assume that we are given $\Omega\in(0,\pi)$, $N\in
{\mathbb{Z}}\,\cup\{+\infty\}$ such that $N\ge 0$. In addition, we are given $s\in{\mathbb{Z}}$ and $q\in\{k\in{\mathbb{Z}}:\  k<s\}\cup\{-\infty\}$.
\par
Let ${\cal T}=\{t\in{\mathbb{Z}}: \ q\le t\le s\}$ if $q>-\infty$ and
${\cal T}=\{t\in{\mathbb{Z}}: \ t\le s\}$ if $q=-\infty$.
\par
Let ${\mathbb{Z}}_N$ be the set of all integers $k$ such that $|k|\le N$ if
$N<+\infty$, and let ${\mathbb{Z}}_N={\mathbb{Z}}$  if
$N=+\infty$.
\par
 For a Hilbert space $H$, we denote by $(\cdot,\cdot)_{H}$ the
corresponding inner product. We denote by $L_2(D)$ the usual Hilbert space of complex valued
square integrable functions $x:D\to{\bf C}$, where $D$ is a interval  in ${\bf R}$.   We denote by $\ell_r$ the set of all
sequences $x=\{x(t)\}_{t\in{\mathbb{Z}}}\subset{\bf C}$, such that
$\|x\|_{\ell_r}=\left(\sum_{t=-\infty}^{\infty}|x(t)|^r\right)^{1/r}<+\infty$
for $r\in[1,\infty)$ or  $\|x\|_{\ell_\infty}=\sup_t|x(t)|<+\infty$
for $r=+\infty$. \par Let $\ell_r^+$ be the set of all sequences
$x\in\ell_r$ such that $x(t)=0$ for $t=-1,-2,-3,...$.

\par
For  $x\in \ell_1$ or $x\in \ell_2$, we denote by $X={\cal Z} x$ the
Z-transform  \begin{eqnarray*} X(z)=\sum_{t=-\infty}^{\infty}x(t)z^{-t},\quad
z\in{\bf C}. \end{eqnarray*} Respectively, the inverse Z-transform  $x={\cal Z}^{-1}X$ is
defined as \begin{eqnarray*} x(t)=\frac{1}{2\pi}\int_{-\pi}^\pi
X\left(e^{i\omega}\right) e^{i\omega t}d\omega, \quad t=0,\pm 1,\pm 2,....\end{eqnarray*}

 If $x\in \ell_2$, then $X|_{\mathbb{T}}$ is defined as an element of
$L_2({\mathbb{T}})$.

Let $\tau\in{\mathbb{Z}}\cup\{+\infty\}$ and $\theta<\tau$; the case where
$\theta=-\infty$  is not excluded. We denote by $\ell_2(\theta,\tau)$ the
Hilbert space of complex valued sequences $\{x(t)\}_{t=\theta}^\tau$
such that
$\|x\|_{\ell_2(\theta,\tau)}=\left(\sum_{t=\theta}^\tau|x(t)|^2\right)^{1/2}<+\infty$.
\par
Let ${\cal Y}_N$ be the Hilbert space of sequences $\{y_k\}_{k=
-N}^{N}\subset{\bf C}$ provided with the $\ell_2$-norm, i.e.,
$\|y\|_{{\cal Y}_N}=\left(\sum_{k\in {\mathbb{Z}}_N }|y_k|^2\right)^{1/2}<+\infty$.

\par
Let ${\mathbb{B}}_\infty$ be the set of all mappings $X:{\mathbb{T}}\to{\bf C}$ such
that $X\left(e^{i\o}\right) \in L_2(-\pi,\pi)$ and $X\left(e^{i\o}\right) =0$ for $|\omega|>\Omega$. We will call the the corresponding processes $x={\cal Z}^{-1}X$
{\em band-limited}.

\par For the case where $N<+\infty$, let ${\mathbb{B}}_N}%oldU_{\O ,N$ be  the set of all $X\in
{\mathbb{B}}_\infty$ such that there exists a sequence
$\{y_k\}_{k=-N}^N\in{\cal Y}_N$ such that $X\left(e^{i\o}\right)
=\sum_{k=-N}^{N}y_ke^{ik\omega\pi/\Omega}{\mathbb{I}}_{\{|\omega|\le\Omega\}}$, where ${\mathbb{I}}$ is
the indicator function.







Consider the Hilbert spaces of sequences ${\cal X}=\ell_2$ and
${\cal X}_-=\ell_2(q,s)$.

 Let $B_N$ be the subset of ${\cal X}_-$ consisting of sequences
$\{x(t)\}_{t\in{\cal T}}$, where $x\in{\cal X}$ are such that $x(t)=({\cal Z}^{-1}
X)(t)$ for $t\in{\cal T}$ for some $X\left(e^{i\o}\right) \in {\mathbb{B}}_N}%oldU_{\O ,N $.

\section{Main results}
\subsubsection*{Uniqueness of the extrapolation for band-limited processes}
\begin{lemma}\label{propU} If  either $q=-\infty$, $N=+\infty$ or $N<+\infty$, $s-q\ge 2N+1$, then  for
  any $x\inB_N $, there exists an
unique  $X\in{\mathbb{B}}_N}%oldU_{\O ,N$ such that $x(t)=({\cal Z}^{-1} X)(t)$ for $t\in{\cal T}$.
\end{lemma}
\par
By Lemma \ref{propU}, the future $\{x(t)\}_{t>s}$ of a
band-limited process $x={\cal Z}^{-1}X$, $X\in{\mathbb{B}}_\infty$,
 is uniquely defined by its  history
$\{x(t),\ t\le s\}$.  This statement represent a reformulation in the deterministic setting
of  the classical Szeg\"o-Kolmogorov Theorem for stationary Gaussian processes
\cite{Sz,Sz1,K,V}. In addition, Lemma \ref{propU} states that the future of  processes
from $B_N $ is uniquely defined by a finite set of historical
values that has at least $2N+1$ elements for any $N<+\infty$.
\begin{remark}\label{rem1}\begin{itemize}
\item[(i)]  If $q>-\infty$ and  $s-q\le  2N+1$, then
 $\{x(t)\}_{t\in{\cal T}}\in B_N $ for any $x\in\ell_2$ (the case where $N=+\infty$ is not excluded). \item[(ii)] If   $s-q< 2N+1$, then there are many
$X\in{\mathbb{B}}_N}%oldU_{\O ,N$ such that $x(t)=({\cal Z}^{-1} X)(t)$  for $t\in{\cal T}$; they form a linear
manifold in $B_N $ (the case where $N=+\infty$, $q>-\infty$ is not excluded).\index{and the following holds:
\subitem(a) If $s-q= 2N+1$, then there is a unique $X\in{\cal U}_{\Omega ,N}$
such that $x(t)=({\cal Z}^{-1} X)(t)$  for $t\in{\cal T}$. \subitem(b) If  $s-q< 2N+1$, then there are many
$X\in{\cal U}_{\Omega ,N}$ such that $x(t)=({\cal Z}^{-1} X)(t)$  for $t\in{\cal T}$; they form a linear
manifold in $B_N $.}
\end{itemize}
\end{remark}
\subsubsection*{Existence and uniqueness of optimal
band-limited approximation} Let $x\in{\cal X}$ be a process. We assume
that the sequence $\{x(t)\}_{t\in{\cal T}}$ represents available
historical data.
\begin{theorem}\label{Th1}\begin{itemize}
\item[(i)] For any $N\le +\infty$, there exists an optimal solution  $\widehat x$
of the minimization problem \begin{eqnarray} \hbox{Minimize}\quad && \sum_{t=q}^s|\widehat
x(t)-x(t)|^2  \quad\hbox{over}\quad \widehat x\in B_N .\label{min} \end{eqnarray}
\item[(ii)]
If either $N=+\infty$, $s=-\infty$ or $N<+\infty$, $s-q\ge
2N+1$, then  the corresponding optimal process $\widehat x$ is uniquely
defined.
\end{itemize}
\end{theorem}
\par
Under the assumptions of Theorem \ref{Th1}(ii), by  Proposition
\ref{propU},  there exists a unique extrapolation of the
band-limited solution $\widehat x$ of problem (\ref{min}) on the future
times  $t>s$. It can
be interpreted  as the optimal forecast (optimal given $\Omega$ and
$N$).
\begin{remark}\label{rem2}
If $N<+\infty$  and  $s-q<2N+1$ then there are
many optimal processes $\widehat x\in B_N$ such that $F(\widehat x,x)=0$; they  form a linear manifold in
$B_N $.
\end{remark}
Up to the end of  this section, we assume that the assumptions of Lemma \ref{propU} and Theorem \ref{Th1}(ii) are satisfied, i..e,   either  $N= +\infty$, $q=-\infty$,
or $N<+\infty$, $s-q\ge 2N+1$.
\subsubsection*{Optimal solution}
Consider the mapping ${\cal Q}: {\cal Y}_N\to B_N $ such that
\begin{eqnarray*} \widehat x(t)=({\cal Q} y)(t)=\frac{\Omega}{\pi}\sum_{k\in
{\mathbb{Z}}_N }y_k {\rm sinc\,}(k\pi+\Omega t).\label{Qs}\end{eqnarray*}  Consider the operator  ${\cal Q}^*:B_N \to {\cal Y}_N$  being adjoint to the operator
${\cal Q}:{\cal Y}_N\toB_N $, i.e., such that
\begin{eqnarray}
({\cal Q}^*x)_k=\frac{\Omega}{\pi}\sum_{t\in{\cal T}}{\rm sinc\,}(k\pi+\Omega t)x(t).
\label{Q*}\end{eqnarray}
By the property of the sinc function, it follows that this convolution maps continuously $\ell_2$ into $\ell_2$. Hence
the operator ${\cal Q}^*$ can be extended as a continuous linear operator ${\cal Q}^*:{\cal X}_-\to {\cal Y}_N$.

Consider the
linear bounded non-negatively defined  Hermitian operator $R={\cal Q}^*{\cal Q}:{\cal Y}_N\to {\cal Y}_N$  defined as
\begin{eqnarray*}
R={\cal Q}^*{\cal Q}.
\end{eqnarray*}
\begin{theorem}
\label{ThP}
\begin{itemize}\item[(i)] The operator $R:{\cal Y}_N\to{\cal Y}_N$ has a bounded inverse
 operator $R^{-1}:{\cal Y}_N\to{\cal Y}_N$.
\item[(ii)] Problem (\ref{min}) has a unique solution \begin{eqnarray}
\widehat x={\cal Q} R^{-1} {\cal Q}^*x.\label{wx}
\end{eqnarray}
\end{itemize}
\end{theorem}
\par
Let us elaborate equation (\ref{wx}).  The optimal  process $\widehat x$ can be expressed as  \begin{eqnarray*} \widehat x(t)=\widehat x
(t,q,s)=\frac{\Omega}{\pi}\sum_{k\in {\mathbb{Z}}_N }\widehat y_k {\rm sinc\,}(k\pi+\Omega t). \label{wxx}\end{eqnarray*}
Here $\widehat y=\{\widehat
y_k\}_{k=-N}^N$ is defined as    \begin{eqnarray} \widehat y=R^{-1}{\cal Q} x.\label{wy}\end{eqnarray} Further, if $N<+\infty$, then the space ${\cal Y}_N$ is finite dimensional, and the operator $R$ can be represented via a matrix
$R=\{R_{km}\}\in {\bf C}^{2N+1,2N+1}$, where $k,m=-N,-N+1,...,N-1,N$.  In this
setting, $(Ry)_k=\sum_{k=-N}^N R_{km}y_m$, and  the components of the matrix $R$  are defined as  \begin{eqnarray*} R_{km}=\index{
\frac{1}{(2\pi)^2}\sum_{t=q}^s\overline{\left(\frac{e^{im\pi+i\Omega t}-
e^{-im\pi-i\Omega t}}{im\pi/\Omega+i t}\right)} \left( \frac{e^{ik\pi+i\Omega
t}- e^{-ik\pi-\Omega t}}{ik\pi/\Omega+i t}\right) \\=\frac{\Omega^2}{(2\pi)^2}
\sum_{t=q}^s\frac{e^{-im\pi-i\Omega t}- e^{imt\Omega t}}{-im\pi-i\Omega t}\cdot
\frac{e^{ik\pi+i\Omega t}- e^{-ik\pi-\Omega t}}{ik\pi+i\Omega t}
\\= \frac{\Omega^2}{(2\pi)^2}\sum_{t=q}^s\frac{-2i\sin(m\pi+\Omega t)}{-im\pi-i\Omega t}\cdot
\frac{2i\sin(k\pi+\Omega t)}{ik\pi+i\Omega t} \\=
\frac{\Omega^2}{\pi^2}\sum_{t=q}^s\frac{\sin(m\pi+\Omega t)}{m\pi+\Omega t}\cdot
\frac{\sin(k\pi+\Omega t)}{k\pi+\Omega t} \\=}
\frac{\Omega^2}{\pi^2}\sum_{j=q}^s{\rm sinc\,}(m\pi+\Omega j)\,{\rm sinc\,}(k\pi+\Omega j)
.\label{R}\end{eqnarray*}
Respectively, the components of the vector  ${\cal Q}^*x=\{({\cal Q}^*x)_k\}_{k=-N}^N$   are defined as
\begin{eqnarray} ({\cal Q}^*x)_{k}= \index{\sum_{t=q}^s\overline{\left(\frac{e^{ik\pi+i\Omega
t}- e^{-ik\pi-i\Omega t}}{ik\pi/\Omega+i t}\right)} x(t)   =\frac{\Omega}{\pi}
\sum_{t=q}^s\frac{\sin(k\pi+\Omega t)}{k\pi+\Omega t}x(t)
 \\=}\frac{\Omega}{\pi} \sum_{j=q}^s{\rm sinc\,}(k\pi+\Omega j)x(j) .
\label{r}\end{eqnarray}


\begin{remark}
 The process $\widehat x(t)$  represents the output of a linear causal smoothing filter.  \index{ for the setting  without memory loss and with dynamically expanding  window of observations, i.e., where $q$ is fixed and  $s$ is running.}   It can be noted that the operators
$R$ and ${\cal Q}$ have to be recalculated for each $s$, and the values $\widehat x(t)=\widehat x_{q,s}(t)$ calculated  for observations $\{x(t),\ s\le t\le q\}$,
 can  be different from the values $\widehat x_{q,s+\tau}(t)$ calculated for the same $t$ using
 the  observations  $\{x(t),\ s\le t\le q+\tau\}$, where  $\tau>0$. Therefore, this filter
 is not time
invariant.
  \index{ However, if we consider a model  with running  $s$ and with fixed memory window,  i.e., where the size $s-q$ is fixed, this filter will be time invariant. To see this, it suffices to assume that  $s$ is fixed and represent the
running  process
$x(t)$ by the process from $\ell_2(q,s)$ via a time shift.}
\end{remark}
\begin{remark} We have  excluded the case where $\Omega=\pi$; this case leads to the trivial solution with $x(-t)=y_t$ for $t\in {\mathbb{Z}}_N $.
\end{remark}

\section{Proofs}
{\em Proof of Lemma \ref{propU}}. Let us consider first the case where $N=+\infty$ and $s-q=-\infty$.
\index{  The  statement (i)  of this proposition for $N=+\infty$ and $q=-\infty$ is known
in principle. For the sake of completeness, we
give below a proof based on Theorem 1 from \cite{D12a}.}  Without a loss of generality, we assume that
 $s=0$. In this case,
${\cal T}=\{t:\ t\le 0\}$.
 It suffices to
prove that if $x(\cdot)\inB_N $ is such that $x(t)=0$ for
$t\in{\cal T}$, then $x(t)=0$ for $t>0$.  By Theorem 1 from \cite{D12a},
processes $x(\cdot)\inB_N $ are weakly predictable in the
following sense: for any $T>0$ , $\varepsilon>0$, and ${\kappa}\in
\ell_\infty(0,T)$, there exists  $\widehat {\kappa}(\cdot)\in
\ell_2(0,+\infty)\cap \ell_\infty(0,+\infty)$ such that
$$
\|y-\widehat y\|_{\ell_2}\le \varepsilon,
$$
where
 \begin{eqnarray*} y(t)\stackrel{{\scriptscriptstyle \Delta}}{=}
\sum_{m=t}^{t+T}{\kappa}(t-m)x(m),\qquad \widehat y(t)\stackrel{{\scriptscriptstyle \Delta}}{=}
\sum^{t}_{m=-\infty}\widehat {\kappa}(t-m)x(m).\label{predict} \end{eqnarray*}
\par
Let us apply this to a process $x(\cdot)\inB_N $ such that
$x(t)=0$ for $t\in{\cal T}$. Let us observe first that \begin{eqnarray} \widehat
y(t)=0\quad \forall t<0. \label{zero} \end{eqnarray} Let $T>0$ be given. Let
us show that $x(t)=0$ if $0\le t\le T$.  Let
$\{{\kappa}_i(\cdot)\}$ be a basis in $\ell_2(-T,0)$. Let
$y_i(t)\stackrel{{\scriptscriptstyle \Delta}}{=}\sum_{m=t}^{t+T} {\kappa}_i(t-m)x(m)$. It follows from
(\ref{zero}) and from the weak predictability \cite{D12a} of $x$  that $y_i(t)=0$ if $t\le 0$.  It
follows that $x(t)=0$ if $t\le T$.
\par
Further, let us apply the proof given above to the process
$x_T(t)=x(t+T)$. Clearly, $x_T(\cdot)\inB_N $ and $x_1(t)=0$
for $t<0$. Similarly, we obtain that $x_T(t)=0$ for all $t\le T$,
i.e., $x(t)=0$ for all $t<2T$. Repeating this procedure $n$ times,
we obtain that $x(t)=0$ for all $t<nT$ for all $n\ge 1$. This
completes the proof of Proposition \ref{propU} for $N=+\infty$ and
$q=-\infty$.

It can be noted that, instead of \cite{D12a}, we could use
predictability of band-limited processes established in \cite{D12b}.

Let us prove the statements (i) for $N<+\infty$, $s-q\ge 2N+1$.  For this case, we use a  different approach
based on non-singularity of special sink matrices established in \cite{LeeF}.  Without a loss of generality, we
assume that $s=N$. \par
Let us consider
first the case when $s-q=2N+1$. It suffices to consider $q=-N$ only; in this
case, the set ${\cal T}=\{t:\ q\le t\le s\}=\{t:\ -N\le t\le N\}$, i.e.,
${\cal T}={\mathbb{Z}}_N $ and it has $2N-1$ elements.
 It suffices to
prove that if $x(\cdot)\inB_N $ is such that $x(t)=0$ for
$t\in{\cal T}$, then $x(t)=0$ for $t>0$. By (\ref{sinc}), we have that
\begin{eqnarray}\sum_{k\in {\mathbb{Z}}_N }a_{t,k}y_k =0,\quad  -N\le t\le N,\label{ay0}\end{eqnarray}
for some set $\{y_k\}$. By Theorem 1(a) from \cite{LeeF}, for $N<+\infty$,  the matrix $\{{\rm sinc\,}(k\pi+\Omega
m)\}_{k,m=-N}^N\in{\bf R}^{2N+1,2N+1}$ is non-singular. Therefore,  linear system
(\ref{ay0}) is a system with a non-singular matrix. Hence $y_k=0$
for all $k$. This proves  Proposition \ref{propU}
(i) for the case where $s-q=2N+1$.

Let us consider the case where $s-q>2N+1$.
In this case, the linear system (\ref{ay0}) has to be considered jointly
with the system \begin{eqnarray}\sum_{k\in {\mathbb{Z}}_N }a_{t,k}y_k =0,\quad  -q\le t<-
N.\label{ay00}\end{eqnarray} Clearly, system (\ref{ay0})-(\ref{ay00}) admits
only a zero solution again. This  completes the proof of Lemma
\ref{propU}. $\Box$

{\em Proof of Remark \ref{rem1}}.
  Assume first that $N<+\infty$ and $s-q=2N+1$. Again, we assume  that $s=N$.
 Since homogeneous
linear system (\ref{ay0}) allows only zero solution, it follows that the
non-homogeneous system \begin{eqnarray}\sum_{k\in {\mathbb{Z}}_N }a_{t,k}y_k =x(t_k),\quad
-N\le t\le N\label{ay1}\end{eqnarray} admits a unique solution $\{y_k\}$ for
any set $\{x(t_k)\}$. Therefore, we proved that
$\{x(t)\}_{t\in{\cal T}}\in B_N $ for any $x\in\ell_2$.  Further, if
$s-q<2N+1$ (including the case where $N=+\infty$), then there are many solutions of (\ref{ay1}), and these
solutions form a linear manifold. This completes the proof of Remark \ref{rem1}. $\Box$
\par
Consider the mapping $\zeta:{\mathbb{B}}_N}%oldU_{\O ,N \to B_N $ such that
$x(t)=(\zeta (X))(t)=({\cal Z}^{-1} X)(t)$ for $t\in{\cal T}$. It is a linear
continuous operator. By Proposition \ref{propU}, it is a bijection.
\par
 {\em
Proof  of Theorem \ref{Th1}.}  Let us prove statement (i). It
suffices to  prove that $B_N $ is a closed linear subspace of
$\ell_2(q,s)$.  In this case,  there exists a unique projection $\widehat
x$ of $\{x(t)\}_{t\in{\cal T}}$ on $B_N $, and the theorem is
proven.
\par
 Clearly, for  any $N\le +\infty$,
the set ${\mathbb{B}}_N}%oldU_{\O ,N$ is a closed linear subspace of $L_2(-\pi,\pi)$.
\par
 Since  the mapping $\zeta:{\mathbb{B}}_N}%oldU_{\O ,N \to B_N $ is continuous, it follows that
the inverse mapping $\zeta^{-1}: B_N \to{\mathbb{B}}_N}%oldU_{\O ,N$ is also
continuous; see, e.g., Corollary in Ch.II.5 \cite{Yosida}, p. 77. Since the
set ${\mathbb{B}}_N}%oldU_{\O ,N$ is a closed linear subspace of $L_2(-\pi,\pi)$, it
follows that $B_N $ is a closed linear subspace of ${\cal X}_-$.
 This completes the proof of
Theorem \ref{Th1} (i). Statement
of Theorem \ref{Th1} (ii) follows  from  Lemma \ref{propU}.  $\Box$
\par
{\em Proof of Remark \ref{rem2}} follows immediately from   Remark \ref{rem1}
and Lemma \ref{propU}.  $\Box$
\par {\em Proof  of Theorem \ref{ThP}.} Let us prove statement (i).
For $N<+\infty$,  statement (i) follows from
Theorem 1(a) from \cite{LeeF} applied to the matrix $\{{\rm sinc\,}(k\pi+\Omega
m)\}_{k,m=-N}^N$.  It can be seen from the following. Let $s=N<+\infty$, $q\le -N$,  $V_0=\{{\rm sinc\,}(k\pi+\Omega
m)\}_{k=-N,m=q}^{N,s}\in{\bf R}^{2N+1,q-s}$, then $R-V_0V_0^\top$ is non-negatively defined. It follows that $R$ is positively defined.
 \par
 Let us prove statement (i) for the general case  where $N\le +\infty$.
We have that ${\mathbb{B}}_N}%oldU_{\O ,N$ is a closed linear subspace of $L_2(-\pi,\pi)$. We established above that the mappings  $\zeta:{\mathbb{B}}_N}%oldU_{\O ,N \to B_N $ and
$\zeta^{-1}: B_N \to {\mathbb{B}}_N}%oldU_{\O ,N$ are continuous.  Since the Fourier series form a basis in $L_2(-\Omega,\Omega)$, it follows that
there is a  continuous bijection between ${\cal Y}_N$ and ${\mathbb{B}}_N}%oldU_{\O ,N$, and the inverse of this bijection is also continuous.  It follows that that the operator  ${\cal Q}:{\cal Y}_{N}\to B_N $ is a continuous
bijection. Respectively, the corresponding
adjoint operator  ${\cal Q}^*:B_N\to {\cal Y}_{N}$ is a  continuous bijection.  Hence the mapping  $R={\cal Q}^*{\cal Q}:{\cal Y}_{N}\to {\cal Y}_{N}$ is also a continuous bijections.  By Corollary in Ch.II.5 \cite{Yosida}, p.77 again,
it follows that
the inverse mapping $R^{-1}: {\cal Y}_{N} \to{\cal Y}_{N}$ is
continuous. This proves statement (i).
\par
Let us prove statement (ii).
Let $\widehat X$ be defined by
\begin{eqnarray*}
\widehat X\left(e^{i\o}\right) =\sum_{k\in
{\mathbb{Z}}_N }y_ke^{ik\omega\pi/\Omega}{\mathbb{I}}_{\{|\omega|\le\Omega\}},
\label{wX}\end{eqnarray*}  where $y=\{y_k\}\in {\cal Y}_N$. Let
$\widehat x={\cal Z}^{-1}\widehat X$. Similarly to the classical sinc representation, we obtain that \begin{eqnarray*} \widehat x(t)=\frac{1}{2\pi}
\int_{-\Omega}^{\Omega}\left(\sum_{k\in {\mathbb{Z}}_N }y_k e^{ik\omega\pi/\Omega}\right)e^{i\omega
t}d\omega\nonumber\\ }% Must be in the body after \begin{documnet=\frac{1}{2\pi}
\sum_{k\in {\mathbb{Z}}_N }y_k\int_{-\Omega}^{\Omega}e^{ik\omega\pi/\Omega+i\omega t}d\omega\nonumber\\
=\frac{1}{2\pi}\sum_{k\in {\mathbb{Z}}_N }y_k \frac{e^{ik\pi+i\Omega t}-
e^{-ik\pi-i\Omega t}}{ik\pi/\Omega+it}\nonumber\\ }% Must be in the body after \begin{documnet=\frac{\Omega}{\pi}\sum_{k\in
{\mathbb{Z}}_N }y_k {\rm sinc\,}(k\pi+\Omega t)=({\cal Q} y)(t).\label{sinc}\end{eqnarray*}
Let the Hermitian form $F:B_N \times
{\cal X}_-\to{\bf R}$ be defined as\begin{eqnarray*} F(\widehat x,x)=\|\widehat x-x\|_{|X_-}^2=\sum_{t=q}^s|\widehat
x(t)-x(t)|^2 . \end{eqnarray*}
Further, let the Hermitian form $G:{\cal Y}_{N}\times {\cal X}_-\to{\bf R}$ be defined as \begin{eqnarray*}
G(y,x)=F({\cal Q} y,x).
\label{formG}\end{eqnarray*} It follows that \begin{eqnarray*}
G(y,x)=\|\widehat x-x\|_{{\cal X}_-}^2 , \quad \widehat x={\cal Q} y.
\label{formGG}\end{eqnarray*}
Clearly, problem (\ref{min}) can be replaced by the minimization problem \begin{eqnarray*}
\hbox{Minimize}\quad && G(y,x)\quad\hbox{over}\quad y\in
{\cal Y}_{N}.\label{minY} \end{eqnarray*}
By the definition, it follows that  \begin{eqnarray*} G(y,x)=({\cal Q} y-x,{\cal Q} y-x)_{{\cal X}_-}= ({\cal Q} y,{\cal Q} y)_{{\cal X}_-}-2{\rm Re\,}({\cal Q} y,x)_{{\cal X}_-}+(
x,x)_{{\cal X}_-}\nonumber\\= ({\cal Q} y,{\cal Q} y)_{{\cal X}_-}-2{\rm Re\,}({\cal Q} y,x)_{{\cal X}_-}+(
x,x)_{{\cal X}_-}\hphantom{xxx}.
\end{eqnarray*}
As was mentioned above, it follows from the properties of the sinc function that the mapping ${\cal Q}^*:\ell_2\to\ell_2$ defined by (\ref{Q*}) is  continuous, and, therefore,
the operator ${\cal Q}^*$ can be extended as a continuous linear operator ${\cal Q}^*:{\cal X}_-\to {\cal Y}_N$.
 It follows that  \begin{eqnarray*}  G(y,x) =(y,Ry)_{{\cal Y}_N}-2{\rm Re\,}(y,{\cal Q}^*x)_{{\cal Y}_N}+(
x,x)_{{\cal X}_-},\label{G}\hphantom{xxx}\end{eqnarray*}
i.e., this is a quadratic form defined on ${\cal Y}_N\times{\cal Y}_N$. By the definitions, the operator $R$ is non-negatively defined, and, by Lemma \ref{propU},
\begin{eqnarray*}
(y,Ry)_{{\cal Y}_N}=({\cal Q} y,{\cal Q} y)_{{\cal X}_-}>0\quad \forall y\neq 0_{{\cal Y}_N}.
\end{eqnarray*} Finally, statement (ii) follows from the invariability of $R$ and the standard properties of the quadratic forms.  This completes the proof of Theorem
\ref{ThP}. $\Box$
 \section{Numerical experiments}
 The experiments show that some eigenvalues of $R$ are
quite close to zero despite the fact that, by Theorem \ref{ThP}, the matrix
$R$ is positively defined. Respectively, the error $\|{\cal Q}^*x-R\widehat y\|_{\ell_2(q,s)}$ for
the MATLAB solution $\widehat y$ of equation  (\ref{wy}) does  not vanish.
In our experiments, we used the Tikhonov regularization
technique  to decrease the error $E$: the matrix $R$ in equation (\ref{wy})  was replaced by
$R_\varepsilon=R+\varepsilon I$, where $I$ is the unit matrix and where $\varepsilon>0$ is
small. In particular, we observed that,  for small $\varepsilon>0$,
\index{$E(\varepsilon)=\|x-\widehat x_\varepsilon\|_{\ell_2(q,s)}< \|x-\widehat
x_\varepsilon\|_{\ell_2(q,s)}$,}
$\|{\cal Q}^*x-R_\varepsilon\widehat y_\varepsilon\|_{\ell_2(q,s)}< \|{\cal Q}^*x-R\widehat
y\|_{\ell_2(q,s)}$, i.e., the approximation  was better and more stable for $\widehat
y_\varepsilon=R_\varepsilon^{-1}{\cal Q}^*x$ calculated for $\varepsilon=0.001$ than for $\widehat y=R^{-1}{\cal Q}^*x$
calculated for $\varepsilon=0$.
In the examples below, we used $\varepsilon=0.001$ and $\varepsilon=0.002$.
\par
In these numerical experiments described below, we have used MATLAB. It appears
that solution of linear system (\ref{wy}) for a given $x$ via build-in MATLAB algorithm
 is more stable than calculation of $R^{-1}$. In particular, calculation of $R_\varepsilon^{-1}$ requires
 larger $\varepsilon$ that solution of (\ref{wy}) for $R=R_\varepsilon$. So far, we found that calculation of inverse matrix  $R_\varepsilon^{-1}$ is feasible for the case of $\varepsilon=0.002$ and $N=200$, i.e., for dimension $(2N+1)\times (2N+1)=401\times 401$.


\par
Figures \ref{fig-1}-\ref{fig-2}  show a  example of  a process $x(t)$ and the
corresponding band-limited process $\widehat x(t)$ approximating $x(t)$
 at times  $t\in \{-25,...,15\}$, i.e., with $q=-25$, $s=15$.
\par
Figure \ref{fig-1}  shows the result for
  $\Omega=0.4$,  $N=15$, $\varepsilon=0.001$, $q=-25$,  and  $s=15$. Figure \ref{fig-2}  shows the result for
  $\Omega=\pi/2$, $N=200$, $\varepsilon=0.002$,  $q=-600$, and  $s=200$.

 
  Figure \ref{fig-2}  shows the result for
 $\Omega=1$.  
The
values of $\widehat x(t)$ for $t>15$  were calculated using the history
$\{x(s)\}_{-25\le s\le 15}$ and can be considered as an optimal  forecast
of $x(t)$.
\par
As was mentioned above, the extrapolation of the process $\widehat x\in
B_N $ to the future times $t>s$ can be interpreted as the
optimal forecast (optimal given $\Omega$ and $N$).
\begin{remark}\label{reme} {\rm
We have used the procedure of replacement $R$ by $R_\varepsilon=R+\varepsilon I$ with
small $\varepsilon>0$ to reduce the error  of calculation of the inverse
matrix for the matrix $R$ that is positively defined but is close to
a singular matrix. It can be noted that the same replacement could
lead to a meaningful setting  for the case when $\varepsilon>0$ is not small.
More precisely, it leads to optimization problem \begin{eqnarray}
\hbox{Minimize}\quad &&
\|\widehat x-x\|^2_{{\cal X}_-}+\varepsilon^2\sum_{k=-N}^N|y_k|^2\nonumber\\&&\quad\hbox{over}\quad y\in
{\cal Y}_{N}, \quad \widehat x={\cal Q} y.\label{minYe} \end{eqnarray} The solution restrains the norm of $y$,
and, respectively, the norm of $\widehat x$, similarly to the setting from \cite{TH}.}
\end{remark}
\begin{figure}[ht
\centerline{\psfig{figure=1n.eps,width=11cm,height=7.0cm}} \caption[]{\sm
Example of discrete time  process $x(t)$ and band-limited discrete time approximation  $\widehat x(t)$
approximating $x(t)$ for $t\in\{-25,..,15\}$, with $\Omega=0.4$, and
$N=15$. The values of $\widehat x(t)$ for $t>15$  were calculated using
$\{x(s)\}_{s=-25}^{15}$ and can be considered as an optimal  forecast
of $x(t)$. } \vspace{0cm}\label{fig-1}\end{figure}
\begin{figure}[ht
\centerline{\psfig{figure=200my.eps,width=15cm,height=7.0cm}} \caption[]{\sm
Example of discrete time process  $x(t)$ and band-limited discrete time approximation $\widehat x(t)$
approximating $x(t)$ for $t\in\{-600,..,200\}$, with $\Omega=\pi/2$, and
$N=200$. The values of $\widehat x(t)$ for $t>200$  were calculated using
$\{x(r)\}_{r=-600}^{200}$ and can be considered as an optimal  forecast
of $x(t)$. } \vspace{0cm}\label{fig-2}\end{figure}
\section{Possible applications and future development}
The  approach suggested in this paper  allows many modifications. We
outline below some possible straightforward modifications as well as
  more challenging problems and possible applications  that we leave for the future research.
\begin{enumerate}
\item For real processes $x(t)$ and $X\left(e^{i\o}\right)={\cal Z} x$, we have that $X\left(e^{-i\omega}\right)=\overline{X\left(e^{i\o}\right)}$.  Therefore,
the optimization problem can be reduced  to optimal selection of the
values of $\widehat X\left(e^{i\o}\right)$ for $\omega\in[0,\Omega]$  only. This would require some
minor adjustments to the solution given above.
\item
Our approach  can be extended on the setting where $x(t)$ is
approximated by a "high frequency" band-limited processes $\widehat x(t)$ such that the
process $\widehat X\left(e^{i\o}\right)$ is supported on $[-\pi,-\pi+\Omega]\cup [\pi-\Omega,\pi]$.
The solution follows immediately form the solution given above  with
$x(t)$ replaced by $(-1)^t x(t)$.  The processes $\widehat x(t)$
are predictable in the sense of weak predictability from Theorem 1 \cite{D12a}.
For the forecasting purposes, it
could be beneficial to use
approximating by  band limited high frequency processes as well.
\item
The set $\{e^{i\omega}, \ \omega\in [-\Omega,\Omega]\}\subset{\mathbb{T}}$ can be replaced by a set that is not
necessary connected, in a setting that is close to one from \cite{F94}. Thus would require a minor modification of the algorithm.
\item It is possible to consider a setting where  sequences of times where the observations of the values  $x(t)$
are missed for some $t$.
\item We have used $2N+1$ terms of the Fourier series expansion  to approximate
$X\left(e^{i\o}\right)$  or $X(i\omega)$ respectively in $L_2(-\Omega,\Omega)$. For the case where
$\Omega$ is close to $\pi$, the corresponding approximating processes
$\widehat x(t)$  decay fast in $t>s$. This decreases the forecasting
horizon.  Possibly, it could be avoided with expansion by another
basis in $L_2(-\Omega,\Omega)$ or $L_2(0,\Omega)$, for instance, such as suggested in \cite{TH}.
\item
 The spaces
$L_2(-\Omega,\Omega)$ and $L_2(0,\Omega)$ can be replaced by  weighted
$L_2$-spaces. This leads to modification of the optimization
problem; the weight will represent the relative importance of the
approximation on different frequencies.
\end{enumerate}
\subsection*{Acknowledgment}
This work  was supported by ARC grant of Australia DP120100928 to the author.
