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\title[]{Positive definite kernels on the set of integers, stability, some properties and applications}

\author{Arnaldo De La Barrera}
\address{Departamento de Matemáticas, Universidad de Pamplona, Pamplona, Colombia}
\email{abarrera1994@gmail.com}

\author{Osmin Ferrer}
\address{Departamento  de Matem\'aticas, Universidad de Sucre, Sincelejo, Colombia}
\email{osmin.ferrer@unisucre.edu.co\newline https://orcid.org/0000-0003-1015-6271}


\author{Jos\'e Sanabria}
\address{Departamento  de Matem\'aticas, Universidad de Sucre, Sincelejo, Colombia}
\email{jose.sanabria@unisucre.edu.co, jesanabri@gmail.com\newline https://orcid.org/0000-0002-9749-4099}


%    General info
\subjclass[2010]{42A82, 60G10}

% 43A35 Positive definite functions on groups, semigroups, etc.
% 42A82     Positive definite functions
% 47D03     Groups and semigroups of linear operators

\date{}

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\keywords{Positive definite kernels, multivariate stochastic processes, Toepliz kernel, Kolgomorov decomposition.}

\begin{abstract}
We define and investigate a class of positive  definite  kernel  so called equivalent-kernel. We formulate and prove an analogous  of Paley-Wiener theorem   in the context of  positive  definite  kernel. The main ingredient in the proof is Kolmogorov decomposition. Finally, some  applications to   stochastic processes are given.
\end{abstract}

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\section* {Introduction}
Positive definite kernels play a prominent role in some applications such as numerical solution of  partial differential
equations, machine learning,  computer graphics, problem moment and probability theory. In the present work we explore some properties of positive definite kernels.  For this kernels one obtains  some similar results
to equivalents bases in Banach spaces  and Riesz bases in Hilbert spaces. An important tool to be used is a version of a classic result due to Kolmogorov, which will be  called a Kolmogorov decomposition of the positive definite kernel $K$ (see \cite{Constantinescu1996}). We will use Kolmogorov decomposition of a  positive definite kernel  to obtain a characterization results of  equivalents  kernels  (see Theorem \ref{equivalent-kernel}). This result is similar to a known result for equivalents bases, Riesz bases and stochastic processes. Using the above, one obtains an analogue Paley-Wiener  Theorem (see   \cite{Young1980}) in the context of positive definite kernels (see Theorem \ref{Paley-Wiener equivalent-kernel}). Finally,  some applications to stochastic processes are given.

\section{  Paley-Wiener Theorem}
Orthonormal bases are very important in Hilbert space theory. There is another less known but also very useful type of bases: the Riesz bases.
This section will be devoted to them. More about these bases can be found in Young's book \cite{Young1980}.


\begin{definition}
A basis in a Hilbert space is a  \textit{Riesz basis} if it is equivalent to an orthonormal basis.
\end{definition}

The fundamental criterium of stability, and historically the first one, is due to Paley  and Wiener
\cite{P-W}. It is based on the known fact that  a linear bounded operator $T$ on a Banach space is invertible if $$\|I - T\| < 1.$$

\begin{theorem}(Paley -Wiener) \label{Paley -Wiener}
Let  $\{x_{n}\}_{n \in \mathbb{N}}$ be a basis in the Banach space $X$,  and   suppose that
$\{y_{n}\}_{n \in \mathbb{N}}$ is a sequence of elements of $X$  such that
\begin{equation*}
\left\| \sum_{n=1}^{N} c_{n}(x_{n}-y_{n}) \right\| \leq \lambda \left\| \sum_{n=1}^{N} c_{n}x_{n}
\right\|,
\end{equation*}
for all $N\in \mathbb{N}$, some constant $\lambda$, with $0 \leq \lambda < 1$ and for any sequence of scalars
$\{c_{n}\}_{n \in \mathbb{N}}$. Then  $\{y_{n}\}_{n \in \mathbb{N}}$ is a basis for $X$ equivalent to $\{x_{n}\}_{n \in \mathbb{N}}$.
\end{theorem}

See  \cite[Theorem 10]{Young1980} for a proof.



\section{Kolmogorov   decomposition theorem}

\subsection{The Hilbert space associated to a positive definite operator valued kernel}$\,$

Let $\{\mathcal{H}_{n}\}_{n \in \mathbb{Z}}$ be a family of Hilbert spaces. An
\textit{operator valued kernel} on $\mathbb{Z}$ to $\{\mathcal{H}_{n}\}_{n \in
\mathbb{Z}}$ is an application $K: \mathbb{Z} \times \mathbb{Z} \rightarrow \bigcup_{m,n \in
\mathbb{Z}} \mathcal{L}(\mathcal{H}_{m}, \mathcal{H}_{n})$ such that
\[
K(n,m) \in\mathcal{L}(\mathcal{H}_{m}, \mathcal{H}_{n})\quad\text{for $n,m \in \mathbb{Z}$.}
\]

In this section and the following one, unless it is otherwise stated,  all the kernels will be operator valued ones.

 A sequence $\{h_{n}\}$ in $\oplus_{n \in \mathbb{Z}} \mathcal{H}_{n}$ is said to have
\textit{finite support}  if $h_{n} = 0$ except for a  finite number of integers $n$.

A kernel $K$ on $\mathbb{Z}$  to $\{\mathcal{H}_{n}\}_{n \in \mathbb{Z}}$ is
a \textit{positive definite kernel} if
\begin{equation*}
\sum_{n,m \in \mathbb{Z}} \langle K(n,m)h_{m},h_{n} \rangle_{\mathcal{H}_{n}} \geq 0,
\end{equation*}
for every sequence $\{h_{n}\}$ in $\oplus_{n \in \mathbb{Z}}\mathcal{H}_{n}$ with finite support.

Let $K$ be a positive definite kernel. Let $\mathcal{F}$ be the linear space of elements
$\bigoplus_{n \in \mathbb{Z}} \mathcal{H}_{n}$ and $\mathcal{F}_{o}$ be the space of elements in
$\mathcal{F}$ with finite support.

Define  $B_{K}: \mathcal{F}_{o} \times \mathcal{F}_{o} \rightarrow \mathbb{C}$ with
\begin{equation}\label{norma-inducida}
B_{K}(f,g) = \sum_{m,n \in \mathbb{Z}} \langle K(n,m)f_{m},g_{n}\rangle_{\mathcal{H}_{n}},
\end{equation}
for $f,g \in \mathcal{F}_{o}$, $f = \{f_{n}\}$, $g = \{g_{n}\}$, $f_{n},g_{n} \in \mathcal{H}_{n}$.

Note that  $B_{K}$ satisfies all the properties of an inner product, except for the fact that
the set
\[
\mathcal{N}_{K}=\{h \in \mathcal{F}_{o} : B_{K}(h,h) = 0\},
\]
could be non-trivial.

According to the Cauchy-Schwarz inequality  $$\mathcal{N}_{K}=\{h \in \mathcal{F}_{o} :
B_{K}(h,g)=0, \textrm{ for all } g \in \mathcal{F}_{o}\},$$ hence $\mathcal{N}_{K}$
is a linear subspace of  $\mathcal{F}_{o}$.

The quotient space $\mathcal{F}_{o}/\mathcal{N}_{K}$ is also a linear subspace. If $[h]$ stands for
the class of the element $h$ in $\mathcal{F}_{o}/\mathcal{N}_{K}$, then the application
\[
\langle[h],[g]\rangle = B_{K}(h,g), \quad h,g \in \mathcal{F}_{o},
\]
is well defined. To prove that $\langle \cdot , \cdot \rangle$ is an inner product on $\mathcal{F}_{o}/\mathcal{N}_{K}$ is straightforward.

The completion of $\mathcal{F}_{o}/\mathcal{N}_{K}$ with respect to the norm induced by this inner product
is a Hilbert  space. It is known as  \textit{the Hilbert space associated to the positive definite kernel $K$} and  it is denoted by $\mathcal{H}_{K}$ . The inner product and the norm of $\mathcal{H}_{K}$ will be represented as $\langle \cdot, \cdot \rangle_{\mathcal{H}_{K}}$ and $\|
\cdot \|_{\mathcal{H}_{K}}$ respectively. This norm will be named as \textit{the norm induced by}
$K$.

\subsection{Kolmogorov Decomposition Theorem}$\,$

The following theorem is a version of the classic result of Kolmogorov (see \cite{Evans1977} for a historical review).
\begin{theorem}[Kolmogorov] \label{Kolmogorov}
Let $K$ be a positive definite kernel. Then there exists  a  Hilbert space  $\mathcal{H}_{K}$    and a map
$V$ defined on $\mathbb{Z}$ such that $V(n) $  belongs to
$\mathcal{L}(\mathcal{H}_{n},\mathcal{H}_{K})$ for each $n \in \mathbb{Z}$ and

\begin{itemize}
\item[(a)] $K(n,m)=V^{\ast}(n)V(m)$ if $n,m\in\mathbb{Z}$.

\item[(b)] $ \displaystyle \mathcal{H}_{K}=\bigvee_{n \in \mathbb{Z}}V(n)\mathcal{H}_{n}$.

\item[(c)] The decomposition is unique in the following sense: if  $\mathcal{H}'$ is another Hilbert space and $V'$ defined on $\mathbb{Z}$ is an application such that $V'(n) \in
\mathcal{L}(\mathcal{H}_{n},\mathcal{H}_{K})$ for each $n \in \mathbb{Z}$ that satisfies  {\rm (a)} and {\rm
(b)}, then there exists a unitary operator  $\Phi: \mathcal{H}_{K} \rightarrow \mathcal{H}'$ such that $\Phi V(n)=V'(n)$ for all $n \in \mathbb{Z}$.
\end{itemize}
\end{theorem}

A proof of this theorem can be found in \cite[Theorem 3.1]{Constantinescu1996}.

\

An application $V$ that satisfies the property  (a) in Theorem   \ref{Kolmogorov} will be called \textit{The Kolmogorov Decomposition of the Kernel $K$}  or simply,  a \textit{Decomposition of
the kernel $K$} (see \cite{Constantinescu1996}). The property  (b) is   referred  to as the \textit{minimality property} of Kolmogorov Decomposition. The meaning of property (c) is that, under the minimality condition  (b),  the Kolmogorov decomposition is essentially  unique.


\section{ Some results for positive definite kernels}

\subsection{Equivalent definite positive kernels }$\,$

Suppose the familiy  of Hilbert spaces  $\{\mathcal{H}_{n}\}_{n \in \mathbb{Z}}$    reduces to a single space, i.e. $\mathcal{H}_{n}=\mathcal{H}   $    for all  $n \in \mathbb{Z}$.


\

In this section some results given in  \cite{Bru-DeLaBar-Dom}  are extended  to the case  of kernel to operator valued .


\begin{definition}\label{def equivalent kernel}

Let  $K_{1}, K_{2} : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$ be two positive definite kernels .

It is said that  $K_{1}$  and  $K_{2}$ are  \textit{equivalent} if there exist two constants  $A,B$ with  $0 < A
\leq B$   such that
\[
A \| [h]_{K_{1}} \|_{\mathcal{H}_{K_{1}}}^{2} \leq \| [h]_{K_{2}} \|_{\mathcal{H}_{K_{2}}}^{2} \leq B \| [h]_{K_{1}}
\|_{\mathcal{H}_{K_{1}}}^{2},
\]
for  $h \in \mathcal{F}_{o}$.
\end{definition}

\begin{remark}\label{norma inducida y clases}
Let  $K: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$  be a positive definite kernel. Let
$h\in \mathcal{F}_{o}$ y $\{h_{n}\}_{n \in \mathbb{Z}}$ a  sequence   in  $\mathcal{H}$ with  finite support.

\

By virtue of  the   definition of norm induced by the kernel $K$  and Kolmogorov decomposition  theorem
it is obtained
\begin{eqnarray*}
\|[h]\|_{\mathcal{H}_{K}}^2&=&\langle [h], [h] \rangle_{\mathcal{H}_{K}}=\sum_{n,m \in \mathbb{Z}} \left\langle K(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}}\mbox{}\\
&=&\sum\limits_{m,n \in \mathbb{Z}}\left\langle V_{K}(n)^{\ast}V_{K}(m)h_{m},h_{n}\right\rangle_{\mathcal{H}}=\left\| \sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n} \right\|_{\mathcal{H}}^{2}.
\end{eqnarray*}
\end{remark}

The following is one of  our results.


\begin{theorem}\label{equivalent-kernel}
Let  $K_{1}, K_{2} : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$ be two positive definite kernels. Then the following conditions are equivalent:

\begin{itemize}

\item[(i)] The kernels  $K_{1}$ y $K_{2}$ are equivalents.

\item[(ii)] There exists a linear bounded bijective  application, with bounded inverse
\[
\Phi : \mathcal{H}_{K_{1}} \to \mathcal{H}_{K_{2}},
\]
such that
\[
\Phi V_{K_{1}}(n) = V_{K_{2}}(n)\quad\text{for all  $n \in \mathbb{Z}$}.
\]

\item[(iii)] There exist two constants  $A,B$ with  $0 < A \leq B$ such that
\[
A \sum_{n,m \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}} \leq
\sum_{n,m \in \mathbb{Z}} \left\langle K_{2}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}} \leq B
\sum_{n,m \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}},
\]
for all sequence with finite support  $\{h_{n}\}_{n \in \mathbb{Z}} \subset \mathcal{H}$.
\end{itemize}

\end{theorem}

\begin{proof}
Let  $V_{K_{1}}$   and  $V_{K_{2}}$ be the Kolmogorov decomposition of the kernels  $K_{1}$, $K_{2}$  and
Let  $\mathcal{H}_{K_{1}}$ and  $\mathcal{H}_{K_{2}}$ the associated Hilbert spaces.

Remark \ref{norma inducida y clases} allows us to write condition (iii) in the following way: there exist two constants  $A$  and $B$ with  $0 < A \leq B$    such that
\[
A \| [h]_{K_{1}} \|_{\mathcal{H}_{K_{1}}}^{2} \leq \| [h]_{K_{2}} \|_{\mathcal{H}_{K_{2}}}^{2} \leq B \| [h]_{K_{1}}
\|_{\mathcal{H}_{K_{1}}}^{2},
\]
for  $h \in \mathcal{F}_{o}$.

Consequently the   conditions  (i) and  (iii) are equivalents .

Next, suppose that condition  (ii)  is true.   Since  $\Phi$  is a linear bounded and invertible operator,  then there exist two constants    $a_o$, $b_o$ with  $0<a_o \leq b_o$   such that
\[
a_o\|f\|_{\mathcal{H}_{K_{1}}} \leq \|\Phi(f)\|_{\mathcal{H}_{K_{2}}} \leq
b_o\|f\|_{\mathcal{H}_{K_{1}}},
\]
for  all  $f\in\mathcal{H}_{K_{1}}$.

Let  $f \in \mathcal{H}_{K_{1}}$   given by
\[
f = \sum_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n},
\]
where  $\{h_{n}\}_{n \in \mathbb{Z}}$ is a sequence in  $\mathcal{H}$  with finite support.

Then
\[
a_o^{2} \left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n} \right\|_{\mathcal{H}_{K_{1}}}^{2}
\leq \left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{2}}(n)h_{n} \right\|_{\mathcal{H}_{K_{2}}}^{2}
\leq b_o^{2} \left\|\sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n}
\right\|_{\mathcal{H}_{K_{1}}}^{2}.
\]
On the other hand, since  $K_{1}$ and  $K_{2}$ are positive definite kernels,  by the Kolmogorov decomposition theorem we have
\begin{equation*}
K_{1}(n,m) = V_{K_{1}}^{\ast}(n)V_{K_{1}}(m), \quad m,n \in \mathbb{Z}
\end{equation*}
and
\begin{equation*}
K_{2}(n,m) = V_{K_{2}}^{\ast}(n)V_{K_{2}}(m), \quad m,n \in \mathbb{Z}.
\end{equation*}

Taking in to account the above expression we have that
\begin{align*}
\left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n} \right\|_{\mathcal{H}_{K_{1}}}^{2} & =
\left\langle \sum\limits_{m \in \mathbb{Z}} V_{K_{1}}(m)h_{m} ,
\sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n} \right\rangle_{\mathcal{H}_{K_{1}}} \\
& = \sum\limits_{m,n \in \mathbb{Z}} \left\langle V_{K_{1}}(n)^{\ast}V_{K_{1}}(m)h_{m},h_{n}\right\rangle_{\mathcal{H}}= \sum\limits_{m,n \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}},
\end{align*}
similarly,
\[
\left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{2}}(n)h_{n} \right\|_{\mathcal{H}_{K_{2}}}^{2}
=\sum\limits_{m,n \in \mathbb{Z}} \left\langle K_{2}(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}.
\]
Thus, choosing   $A=a_o^{2}$  and  $B=b_o^{2}$ we have
\[
A \sum\limits_{m,n \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}
\leq \sum\limits_{m,n \in \mathbb{Z}}\left\langle K_{2}(n,m)h_{m},h_{n}
\right\rangle_{\mathcal{H}} \leq B \sum\limits_{m,n \in \mathbb{Z}}\left\langle
K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}},
\]
where  $\{h_{n}\}_{n \in \mathbb{Z}}$  is a sequence in   $\mathcal{H}$ with finite support .

Now, let us suppose  that condition  (iii)  is valid.

The application  $\Phi_{o} : \mathcal{F}_{o,K_{1}} \to \mathcal{F}_{o,K_{2}}$  is defined as follows
\[
\Phi_{o} \left( \sum_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n} \right)
= \sum_{n \in \mathbb{Z}} V_{K_{2}}(n)h_{n},
\]
where  $\{h_{n}\}_{n \in \mathbb{Z}}$ is a sequence in  $\mathcal{H}$  with finite support.  It is not hard to prove that  $\Phi_{o}$  is a linear operator.

In what follows we will proof that  $\Phi_{o}$  is a bounded above and bounded below operator.

By  the Kolmogorov decomposition  theorem we obtain
\[
\sum\limits_{m,n \in \mathbb{Z}} \left\langle
K_{2}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}} = \sum\limits_{m,n \in \mathbb{Z}}\left\langle
V_{K_{2}}(n)^{\ast}V_{K_{2}}(m)h_{m},h_{n}\right\rangle_{\mathcal{H}}.
\]
Taking  into account the above result and the way that the operator  $\Phi_{o}$  was defined we arrive to the next result
\begin{eqnarray*}
\sum\limits_{m,n \in \mathbb{Z}}\left\langle K_{2}(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}&=&\left\langle \sum\limits_{m \in \mathbb{Z}} V_{K_{2}}(m)h_{m} ,\sum\limits_{n \in \mathbb{Z}}
 V_{K_{2}}(n)h_{n} \right\rangle_{\mathcal{H}_{K_{2}}}\mbox{}\\
&=&\left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{2}}(n)h_{n} \right\|_{\mathcal{H}_{K_{2}}}^{2}=\left\|\Phi_{o} \left(\sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n} \right) \right\|_{\mathcal{H}_{K_{2}}}^{2}.
\end{eqnarray*}

In a similar way we have
\[
\sum\limits_{m,n \in \mathbb{Z} } \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}}
= \left\|\sum\limits_{n \in \mathbb{Z}} V_{K_{1}}h_{n} \right\|_{\mathcal{H}_{K_{1}}}^{2}.
\]
By (iii),
\[
A \left\|\sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n} \right\|_{\mathcal{H}_{K_{1}}}^{2} \leq
\left\|\Phi_{o}\left(\sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n} \right)
\right\|_{\mathcal{H}_{K_{2}}}^{2} \leq B \left\|\sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n}
\right\|_{\mathcal{H}_{K_{1}}}^{2}.
\]
The last chain of inequalities shows us that $\Phi_{o}$  is a bounded above and bounded below operator.  Even more the domain and the range of $\Phi_{o}$  are dense in the spaces   $\mathcal{H}_{K_{1}}$   and    $\mathcal{H}_{K_{2}}$ respectively. Then this operator can be extended to a bounded operator with bounded inverse say $\Phi: \mathcal{H}_{K_{1}} \rightarrow \mathcal{H}_{K_{2}}$. By construction
\[
\Phi V_{K_{1}}(n) = V_{K_{2}}(n) \quad \text{for all } \quad n\in \mathbb{Z}.
\]
\end{proof}

Theorem  \ref{equivalent-kernel}  has similarities with results referring to equivalent basic sequences in Banach spaces, for more details   on the topic  (see  \cite{Lind-Tzaf, Carothers2005}).

Our next stability result for positive definite kernels is similar to a stability theorem for equivalent bases due to Paley-Wiener (see \cite[Theorem  10]{Young1980}).
	
In  first place we will fix the notation. Given two positive definite kernels
$K:  \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$ and  $K_{1}: \mathbb{Z} \times
\mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$,  let  $V_{K}$   and  $V_{K_{1}}$ the
Kolmogorov decompositions of  $K$ and  $K_{1}$  respectively and let  $\mathcal{H}_{K}$ and
$\mathcal{H}_{K_{1}}$   the induced  Hilbert  spaces.

\begin{theorem}\label{Paley-Wiener equivalent-kernel}
Let  $K: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$  and $K_{1}: \mathbb{Z}\times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$ be two positive definite kernels. If
$V_{K_{1}}(n)\in \mathcal{L}(\mathcal{H},\mathcal{H}_{K})$  for all  $n \in \mathbb{Z}$ and satisfies
\begin{equation*}\label{desigualdad PW-KEQUIVA}
\left\| \sum\limits_{n \in \mathbb{Z}}(V_{K}(n)-V_{K_{1}}(n))h_{n}
\right\|_{\mathcal{H}_{K}} \leq \lambda\left\| \sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n}
\right\|_{\mathcal{H}_{K}},
\end{equation*}
for any sequence with finite support $\{h_{n}\}_{n \in \mathbb{Z}} \subset
\mathcal{H}$, where  $\lambda \in (0,1)$, then  $K_{1}$  is  equivalent to  $K$.
\end{theorem}

\begin{proof}
Let us define the operator  $T: \mathcal{H}_{K} \rightarrow \mathcal{H}_{K}$   as follows
\begin{equation*}
T\left(\sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n} \right) = \sum\limits_{n \in
\mathbb{Z}}(V_{K}(n)-V_{K_{1}}(n))h_{n},
\end{equation*}
where   $\{h_{n}\}_{n \in \mathbb{Z}}$   is a sequence in  $\mathcal{H}$   with finite support.

By hypothesis  $T$   is well defined and it is a linear operator. From the definition of  $T$   and by hypothesis we have.
\[
\left\| T\left(\sum\limits_{n \in
	\mathbb{Z}}V_{K}(n)h_{n} \right) \right\|_{\mathcal{H}_{K}}^{2} \leq \lambda^{2}
\left\|\sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n} \right\|_{\mathcal{H}_{K}}^{2}.
\]
Hence,  $T$   is a bounded operator and moreover
\[
\|T \| \leq \left| \lambda\right| <1.
\]
Next,  let us consider the operator $I-T: \ \mathcal{H}_{K} \rightarrow \mathcal{H}_{K}$, as usual $I: \ \mathcal{H}_{K} \rightarrow \mathcal{H}_{K}$  is the identity operator .

Since   $\| T \| <1$,   by a well known functional analysis Theorem, $I - T$    is an invertible bounded linear operator. Moreover,
\begin{align*}
 (I-T) \left( \sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n}\right)
& = \sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n}
- T\left( \sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n} \right) \\
& = \sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n}
- \left( \sum\limits_{n \in \mathbb{Z}} (V_{K}(n)-V_{K_{1}}(n))h_{n} \right)= \sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n}.
\end{align*}
From the above,  it follows that there are positive constants  $m$ and $M$  with  $m\leq M$
such that
\begin{align*}
m \left\|\sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n} \right\|_{\mathcal{H}_{K}} &
\leq\left\|\left(I-T\right)\left(\sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n} \right)
\right\|_{\mathcal{H}_{K}} \\& =\left\|\sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n}
\right\|_{\mathcal{H}_{K}}\leq M \left\|\sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n}
\right\|_{\mathcal{H}_{K}}.
\end{align*}
By  Remark  \ref{norma inducida y clases}
\[
\left\|\sum\limits_{n \in \mathbb{Z}} V_{K}(n)h_{n} \right\|_{\mathcal{H}_{K}}^{2}
= \sum\limits_{m,n \in \mathbb{Z}} \left\langle
K(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}.
\]
By hypothesis  $V_{K_{1}}(n)\in \mathcal{L}(\mathcal{H},\mathcal{H}_{K})$ for all  $n \in
\mathbb{Z}$,  thus  $V_{K_{1}}(n)h_{n} \in \mathcal{H}_{K}$. Then
\begin{align*}
\sum_{n,m \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}}
& =\sum\limits_{m,n \in \mathbb{Z}}\left\langle V_{K_{1}}(n)^{\ast}V_{K_{1}}(m)h_{m},h_{n}\right\rangle_{\mathcal{H}}\\
& =\sum\limits_{m,n \in \mathbb{Z}}\left\langle V_{K_{1}}(m)h_{m},V_{K_{1}}(n)h_{n}\right\rangle_{\mathcal{H}_{K}} =\left\| \sum\limits_{n \in \mathbb{Z}} V_{K_{1}}(n)h_{n} \right\|_{\mathcal{H}_{K}}^{2}.
\end{align*}
Replacing these expressions in the above inequalities,  we derive the existence of positive constants   $A$  and $B$ with  $A\leq B$ such that
\[
A \sum\limits_{m,n \in \mathbb{Z}} \left\langle K(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}
\leq \sum\limits_{m,n \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}
\leq B \sum\limits_{m,n \in \mathbb{Z}} \left\langle K(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}},
\]
for all sequences $\{h_{n}\}_{n \in \mathbb{Z}}$ in $\mathcal{H}$ with finite support .

Applying Theorem \ref{equivalent-kernel}, it follows that  $K_{1}$  is  equivalent to $K$.
\end{proof}










\section{Applications to Stochastic Processes}


\subsection{Multivariate Stochastic Processes}$\,$

In this section it will be used the decomposition of the covariance Kernels  of the stochastic processes (see \cite{Constantinescu1996}, Section 1,  Chapter 6).

Let $(\Omega,{F},P)$ be a probability space, where  ${F}$ is a  $\sigma$-algebra of
subsets of $\Omega $ and  $P$ is a probability measure on  $F$. A  \textit{stochastic variable} is a function
$x: \Omega \rightarrow \mathbb{C}$, which is measurable with respect to the   $\sigma$-algebra $F$. \textit{A stochastic process} is a family $\{x_{n}\}_{n \in
\mathbb{Z}}$ of stochastic variables. Let
  $L^{2}(P)$ be the Hilbert space of the measurable functions from ${F}$  to $\Omega$ with integrable square, this is,
\[
L^{2}(P) = \left\{ x: \Omega \rightarrow \mathbb{C}: \, x \textrm{ is a measurable function and} \int_{\Omega} | x(\omega) |^{2} dP(\omega) < +\infty \right\}
\]
equipped with the inner product
\[
\langle x,y  \rangle_{L^{2}(P)} = \int_{\Omega} x(\omega)\overline{y(\omega)} dP(\omega).
\]
From here on, only stochastic processes with variables in $L^{2}(P)$ will be considered.

The mean-value variable  is defined by
\[
m_{n} = E(x_{n}) = \int_{\Omega} x_{n}(\omega )dP(\omega)
\]
and it is convenient to assume that  $m_{n} = 0$ for all $n \in \mathbb{Z}$. The correlation of the stochastic process  $\{x_{n}\}_{n \in
\mathbb{Z}}$ is given by
\[
K(m,n) = K_{mn} =\int_{\Omega} x_{n}(\omega)\overline{x_{m}(\omega)} dP(\omega)= \langle x_{n},x_{m} \rangle_{L^{2}(P)}.
\]
for all  $m,n \in \mathbb{Z}$.

It is straightforward that the correlation kernel of this process is a positive definite kernel.   In fact
\begin{align*}
\sum_{i,j = m}^{n} K_{ij}\lambda _{j}\overline{\lambda}_{i}&= \sum_{i,j = m}^{n} \langle
x_{j},x_{i} \rangle_{L^{2}(P)} \lambda _{j}\overline{\lambda} _{i} = \sum_{i,j = m}^{n} \langle \lambda
_{j}x_{j},\lambda _{i}x_{i} \rangle_{L^{2}(P)}= \left\| \sum_{j = m}^{n} \lambda _{j}x_{j}\right\|_{L^{2}(P)} ^{2}
\geq 0,
\end{align*}
for all $m,n \in \mathbb{Z}, m \leq n,$  and $\lambda _{k}\in \mathbb{C}$, where $k = m,m+1,...,n$.

A stochastic process $\{x_{n}\}_{n \in \mathbb{Z}}$  is  said to be \textit{stationary (in a wide sense)} if its  correlation kernel is a Toeplitz kernel, that is
\[
K(m,n)=K_{n-m}   \quad \text{for all}   \quad  m,n \in \mathbb{Z}.
\]
In this case it can be used the Naimark Decomposition  Theorem  in order to associate the stationary stochastic process    $\{x_{n}\}_{n \in\mathbb{Z}}$ with the Hilbert space  $\mathcal{H}_{K}$, the unitary operator $S\in L(\mathcal{H}_{K})$  and the operator $Q\in L(\mathbb{C}, \mathcal{H}_{K})$  such that
\[
K_{n}=Q^{\ast}S^{n}Q,  \quad    n \in\mathbb{Z}.
\]
The geometric settings  for the prediction problem can be extended  in order to deal with the multivariate case too. Let notice that  a random variable  $x_{n}: \Omega \rightarrow \mathbb{C}$, of a stochastic process $\{ x_{n} \}_{n \in \mathbb{Z}} \subset L^{2}(P)$, can be interpreted as an operator  from $\mathbb{C}$ to $L^{2}(P)$ defining
$\widetilde{x}_{n}: \mathbb{C} \rightarrow L^{2}(P)$ as
\[
\widetilde{x}_{n} (\lambda) = \lambda x_{n},
\]
and the elements of the correlation kernel of the process can be calculated according to the rule
\[
K(m,n)=(\widetilde{x}_{m})^{\ast}\widetilde{x}_{n}.
\]
Also, it must be noticed that many stochastic processes  have the same correlation kernel.  Having this in mind it is convenient to adopt the following terminology. The main object used to describe a \textit{multivariate process } will be its correlation kernel $K$  which is supposed to be
positive definite and  $K(m,n) \in \mathcal{L}(\mathcal{H}_{n},\mathcal{H}_{m})$ for all  $m,n \in \mathbb{Z}$, where
$\textbf{H} = \{\mathcal{H}_{n}\}_{n \in \mathbb{Z}}$ is a family of Hilbert spaces.

\begin{definition}
A pair $[\mathcal{K},X]$, where $\mathcal{K}$ is a Hilbert space and $X=\{X_{n}\}_{n \in\mathbb{Z}}$  is a family of operators  $X_{n}$ in $\mathcal{L}(\mathcal{H}_{n},\mathcal{K})$, is called a \textit{geometric model of the multivariate process}  with correlation kernel $K$, if
\[
K(m,n)=X^{\ast}_{m}X_{n}.
\]
\end{definition}

The  Kolmogorov Decomposition Theorem shows that given a positive definite kernel $K$, there exists a geometric model of the multivariate process  with correlation kernel $K$. If $[\mathcal{K},X]$ is the geometric model of the multivariate process with covariance kernel $K$  then $\mathcal{H}_{X}$ will be the  subspace of $\mathcal{K}$ generated for this model, that is,
\begin{equation}
\mathcal{H}_{X}=\bigvee_{n \in \mathbb{Z}}X_{n}\mathcal{H}_{n}.
\end{equation}
If $[\mathcal{K}^{\prime},X^{\prime}]$  is another geometric model of the same process, then
the Kolmogorov Decomposition Theorem guarantees the existence of an unitary operator  $\Phi: \mathcal{H}_{X} \rightarrow \mathcal{H}_{X^{\prime}}$ such that $\Phi X_{n}=X^{\prime}_{n}$ for all
  $n \in \mathbb{Z}$. This means that the geometry of the process is essentially determined  by the choise  of a geometric model such that
\begin{equation}
\mathcal{K}=\bigvee_{n \in \mathbb{Z}}X_{n}\mathcal{H}_{n}.
\end{equation}

\subsection{Equivalent Multivariate Stochastic Processes }$\,$

From here on,   $\mathcal{H}_{n} = \mathcal{H}$ for all $n \in \mathbb{Z}$ and the covariance kernels of the processes will be positive definite.

\begin{theorem}[Isomorphism]\label{isomorfismo  operadores}
 Let  $[\mathcal{W},X]$ be the geometric model of a multivariate   process and  let
 $K: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$ be  the kernel of covariance associated with the process.   Then there exists an unit operator  $\Phi :\mathcal{H}_{K}\rightarrow \mathcal{H}_{X} $  such that
\begin{equation*}
\Phi V_{K}(n)=X_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}.
\end{equation*}
 \end{theorem}

\begin{proof}
Let  $[\mathcal{W},X]$, $X=\{X_{n}\}_{n \in\mathbb{Z}}$ be a geometric model of a multivariate process and $K: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathcal{L}(\mathcal{H})$  be the kernel of covariance associated with the process.

It follows that the covariance kernel and the space generated  by the process is given by
\begin{equation*}
K(n,m)=X^{\ast}_{n}X_{m} \quad  \text{and}  \quad  \mathcal{H}_{X}=\bigvee_{n \in \mathbb{Z}}X_{n}\mathcal{H}.
\end{equation*}
On the other hand, since  $K$   is a positive definite kernel one more time by the Kolmogorov decomposition theorem there exists a Hilbert space    $\mathcal{H}_{K}$  and an application   $V_{K}(n)\in \mathcal{L}(\mathcal{H}, \mathcal{H}_{K}) $ for all  $n \in\mathbb{Z}$  such that
\begin{equation*}
K(n,m)=V^{\ast}_{K}(n)V_{K}(m)  \quad  \text{and}  \quad  \mathcal{H}_{K}=\bigvee_{n \in \mathbb{Z}}V_{K}(n)\mathcal{H}.
\end{equation*}
Let us define the application
$\Phi :\mathcal{H}_{K}\rightarrow \mathcal{H}_{X} $ in the following way
\begin{equation*}
\Phi\left(\sum\limits_{n \in \mathbb{Z}}V_{K}(n) h_{n} \right) =  \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n},
\end{equation*}
where  $\{h_{n}\}_{n \in \mathbb{Z}}$  is a sequence  with finite support in $\mathcal{H}$.

Then we have

\begin{align*}
\left\|\Phi\left(\sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n} \right) \right\|_{\mathcal{H}_{X}}^{2} &
 =\left\| \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}\right\|_{\mathcal{H}_{X}}^{2}=\sum\limits_{m,n \in \mathbb{Z}} \left\langle X_{m}h_{m},X_{n}h_{n}\right\rangle_{\mathcal{H}_{X}} \\&
 = \sum\limits_{m,n \in \mathbb{Z}} \left\langle K(n,m)h_{m},h_{n}\right\rangle_{\mathcal{H}}=\sum\limits_{m,n \in \mathbb{Z}} \left\langle V^{\ast}_{K}(n)V_{K}(m) h_{m},h_{n}\right\rangle_{\mathcal{H}} \\&
 = \sum\limits_{m,n \in \mathbb{Z}} \left\langle V_{K}(m) h_{m},V_{K}(n)h_{n}\right\rangle_{K} =\left\|\sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n} \right\|_{\mathcal{H}_{K}}^{2},
\end{align*}
all of this show us that the application   $\Phi$   can be extended by continuity to an unit operator from   $\mathcal{H}_{K}$   over    $\mathcal{H}_{X}$
and moreover  $\Phi V_{K}(n)=X_{n}$   for  all  $n \in \mathbb{Z}$.
\end{proof}

\begin{definition}
Two geometric models of multivariate processes   $[\mathcal{K},X]$ and $[\mathcal{L},Y]$    are  said to be \textit{ equivalent  },
 if  $\text{dim}\,(\mathcal{H}_{X}) = \text{dim}\,(\mathcal{H}_{Y})$ and there are two constants  $A,B$  with  $0 < A
\leq B$ such that
\[
A \left\| \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}\right\|_{\mathcal{H}_{X}}^{2}  \leq \left\| \sum\limits_{n \in \mathbb{Z}}Y_{n}h_{n}\right\|_{\mathcal{H}_{Y}}^{2}  \leq B  \left\| \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}\right\|_{\mathcal{H}_{X}}^{2}  ,
\]
where  $\{h_{n}\}_{n \in \mathbb{Z}}$ is a sequence in $\mathcal{H}$  with finite support.
\end{definition}

By Theorem \ref{isomorfismo operadores}  and definitions we have the following.

\begin{proposition}\label{def procesos equivalent}
Let    $[\mathcal{W},X]$ and  $[\mathcal{W}_{1},Y]$  be two geometric  model of multivariate process and let     $K_{1}$  and   $K_{2}$   be two kernels of covariance associated with the processes. Then  $K_{1}$  and   $K_{2}$  are  \textit{equivalent kernels }  if and only if   $X = \{ X_{n} \}_{n \in \mathbb{Z}}$  and   $Y =\{ Y_{n} \}_{n \in \mathbb{Z}}$  are equivalent processes.
\end{proposition}

As an application we give the proof of the results obtained in  \cite{DelabarreraLoraFerrer2017}.

\begin{theorem}\label{EQUIVALENT-PROCESS}
Let $[\mathcal{K},X]$  and  $[\mathcal{L},Y]$ be  two geometric models of multivariate processes.   The following conditions are equivalent:
\begin{itemize}
\item[(i)]   The models of the multivariate processes  $[\mathcal{K},X]$ and  $[\mathcal{L},Y]$    are equivalent.
\item[(ii)] There is a bijective bounded linear application with bounded inverse
$\psi : \mathcal{H}_{X} \to \mathcal{H}_{Y}$ such that
\[
\psi X_{n} = Y_{n} \quad \text{for all $n \in \mathbb{Z}.$}
\]
\item[(iii)] There exist two constants $A,B$ with $0 < A \leq B$ such that
\[
A \left\| \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}\right\|_{\mathcal{H}_{X}}^{2}  \leq \left\| \sum\limits_{n \in \mathbb{Z}}Y_{n}h_{n}\right\|_{\mathcal{H}_{Y}}^{2}  \leq B  \left\| \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}\right\|_{\mathcal{H}_{X}}^{2},
\]
for each sequence with finite support $\{h_{n}\}_{n \in \mathbb{Z}} \subset \mathcal{H}$.
\end{itemize}
\end{theorem}


\begin{proof}
The equivalence between    (i) and   (iii)  follows by definition. Next,  we are going to show that  (i) implies  (ii) to this end let us assume that   $X = \{ X_{n} \}_{n \in \mathbb{Z}}$  and   $Y =\{ Y_{n} \}_{n \in \mathbb{Z}}$  are equivalent processes
let  $K_{1}$  and  $K_{2}$  be the kernels of covariance associated with the processes   $X = \{ X_{n} \}_{n \in \mathbb{Z}}$  and    $Y =\{ Y_{n} \}_{n \in \mathbb{Z}}$ respectively. Since  $X = \{ X_{n} \}_{n \in \mathbb{Z}}$ and   $Y =\{ Y_{n} \}_{n \in \mathbb{Z}}$   are equivalent,  then  by proposition    \ref{def procesos equivalent} we concluded that  $K_{1}$  and   $K_{2}$ are   equivalent kernels. By Theorem  \ref{equivalent-kernel},  there exists a biyective bounded linear application linear with bounded inverse
$\Phi : \mathcal{H}_{K_{1}} \to \mathcal{H}_{K_{2}}$   such that
\[
\Phi V_{K_{1}}(n) = V_{K_{2}}(n) \quad \text{for all  } n \in \mathbb{Z}.
\]
Let  us  consider the operators  $\phi_{1} :\mathcal{H}_{K_{1}}\rightarrow \mathcal{H}_{X} $   such that
 \begin{equation*}
 \phi_{1} V_{K_{1}}(n)=X_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}
\end{equation*}
and $\phi_{2} :\mathcal{H}_{K_{2}}\rightarrow \mathcal{H}_{Y}$  such that
\begin{equation*}
 \phi_{2} V_{K_{2}}(n)=Y_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}.
\end{equation*}
From the above it  follows  that
\begin{equation*}
 \phi^{-1}_{2}\Phi \phi^{-1}_{1}X_{n}=Y_{n}    \quad \text{for all }  \quad  n \in \mathbb{Z}.
\end{equation*}

Now suppose that  (ii) holds then there is a bijective  bounded linear application with bounded inverse $\psi : \mathcal{H}_{X} \to \mathcal{H}_{Y}$   such that
\[
\psi X_{n} = Y_{n} \quad \text{for all $n \in \mathbb{Z}.$}
\]
Let  $K_{1}$ and $K_{2}$  be two kernels of covariance associated with the processes $X = \{ X_{n} \}_{n \in \mathbb{Z}}$  and  $Y =\{ Y_{n} \}_{n \in \mathbb{Z}}$, respectively .

Let  us  consider the operators $\phi_{1} :\mathcal{H}_{K_{1}}\rightarrow \mathcal{H}_{X} $  such that
 \begin{equation*}
 \phi_{1} V_{K_{1}}(n)=X_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}
\end{equation*}
and $\phi_{2} :\mathcal{H}_{K_{2}}\rightarrow \mathcal{H}_{Y} $  such that
 \begin{equation*}
 \phi_{2} V_{K_{2}}(n)=Y_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}.
\end{equation*}
From the above it  follows that
\begin{equation*}
 \phi^{-1}_{2}\psi \phi_{1}V_{K_{1}}(n)=V_{K_{2}}(n)    \quad \text{for all  }  \quad  n \in \mathbb{Z}.
\end{equation*}
By Theorem  \ref{equivalent-kernel},  we obtain     $\text{dim}\,(\mathcal{H}_{K_{1}}) = \text{dim}\,(\mathcal{H}_{K_{2}})$ and there exist two positive constants  $A,B$, $A \leq B$   such that
\[
A \sum_{n,m \in \mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}} \leq
\sum_{n,m \in \mathbb{Z}} \left\langle K_{2}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}} \leq  B \sum_{n,m \in
\mathbb{Z}} \left\langle K_{1}(n,m)h_{m},h_{n} \right\rangle_{\mathcal{H}},
\]
where  $\{h_{n}\}_{n \in \mathbb{Z}}$ is a sequence in  $\mathcal{H}$ with finite  support.

The result comes up from the fact that $K_{1}(m,n)=X^{\ast}_{m}X_{n}$  and $K_{2}(m,n)=Y^{\ast}_{m}Y_{n}$.
\end{proof}

In the  multivariate stochastic processes setting it is possible to obtain a result similar to that of the theorem on stability (see Theorem  \ref{Paley -Wiener}).

The following is our result about stability of multivariate stochastic processes.

\begin{theorem}\label{Paley-Wiener EQUIVALENT-PROCESS}
Let  $[\mathcal{W},Y]$  be a geometrical  model of a multivariate stochastic process,     $\mathcal{H}_{Y}$ the subspace generated by the process, and suppose  $X_{n}\in \mathcal{L}(\mathcal{H},\mathcal{H}_{Y})$  for all $n \in \mathbb{Z}$     such that
\begin{equation}\label{desigualdad1 PW-EQUIVAPROCESS}
\left\| \sum\limits_{n \in \mathbb{Z}}(Y_{n}-X_{n})h_{n} \right\|_{\mathcal{H}_{Y}}
\leq \delta\left\| \sum\limits_{n \in \mathbb{Z}}Y_{n}h_{n} \right\|_{\mathcal{H}_{Y}},
\end{equation}
for some constant  $\delta$, $0<\delta<1  $,  and any sequence  $\{h_{n}\}_{n \in \mathbb{Z}}$ in $\mathcal{H}$ with finite support.
 Then  the geometric  model of the multivariate process $[\mathcal{K},X]$   is equivalent to $[\mathcal{W},Y]$.
\end{theorem}

\begin{proof}
Let $K$  and    $K_{1}$ be two kernels of covariance associated with the processes $Y = \{ Y_{n} \}_{n \in \mathbb{Z}}$  and   $X =\{ X_{n} \}_{n \in \mathbb{Z}}$, respectively.

Let us consider the operators $\Phi_{1} :\mathcal{H}_{K}\rightarrow \mathcal{H}_{Y} $  such that
 \begin{equation*}
 \Phi_{1} V_{K}(n)=Y_{n}    \quad \text{for all }  \quad  n \in \mathbb{Z}
\end{equation*}
and $\Phi_{2} :\mathcal{H}_{K_{1}}\rightarrow \mathcal{H}_{X} $  such that
\begin{equation*}
 \Phi_{2} V_{K_{1}}(n)=X_{n}    \quad \text{for all  }  \quad  n \in \mathbb{Z}.
\end{equation*}
From the above and hypothesis we have
\[
\mathcal{H}_{K_{1}}\subset \mathcal{H}_{K}  \quad  \text{and} \quad   \Phi_{2}\left(\sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n} \right)= \sum\limits_{n \in \mathbb{Z}}X_{n}h_{n}  =\Phi_{1}\left(\sum\limits_{n \in \mathbb{Z}}V_{K_{1}}(n)h_{n} \right).
\]
Then
\begin{align*}
\left\| \sum\limits_{n \in \mathbb{Z}}(V_{K}(n)-V_{K_{1}}(n))h_{n} \right\|_{\mathcal{H}_{K}} &
= \left\| \Phi_{1}\sum\limits_{n \in \mathbb{Z}}(V_{K}(n)-V_{K_{1}}(n))h_{n} \right\|_{\mathcal{H}_{Y}}\\ &
=\left\| \sum\limits_{n \in \mathbb{Z}}(\Phi_{1}V_{K}(n)-\Phi_{2}V_{K_{1}}(n))h_{n} \right\|_{\mathcal{H}_{Y}}=  \left\| \sum\limits_{n \in \mathbb{Z}}(Y_{n}-X_{n})h_{n} \right\|_{\mathcal{H}_{Y}}\\ &
\leq \delta\left\| \sum\limits_{n \in \mathbb{Z}}Y_{n}h_{n} \right\|_{\mathcal{H}_{Y}}=\delta\left\| \sum\limits_{n \in \mathbb{Z}}\Phi_{1}V_{K}(n)h_{n} \right\|_{\mathcal{H}_{Y}}  \\ &
=\delta\left\| \sum\limits_{n \in \mathbb{Z}}V_{K}(n)h_{n} \right\|_{\mathcal{H}_{K}},
\end{align*}
for any sequence  $\{h_{n}\}_{n \in \mathbb{Z}}$ in  $\mathcal{H}$ with finite support.

Finally, by Theorem \ref{Paley-Wiener equivalent-kernel} it follows that
  $K_{1}$  and    $K$  are equivalent kernels. Therefore  $X=\{X_{n}\}_{n \in\mathbb{Z}}$ is equivalent to $Y=\{Y_{n}\}_{n \in\mathbb{Z}}$.
\end{proof}


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