\documentstyle{SibMatJ}
%\TestXML
\topmatter

  \Author           Rani
    \Initial        R.
    \Sign           Rekha Rani
    \Email          rekha222017\@cuh.ac.in
    \AffilRef       1
  \endAuthor

  \Author           Jahan
    \Initial        S.
    \Sign           Shah Jahan
    \Email          shahjahan\@cuh.ac.in
    \AffilRef       1
    \Corresponding
  \endAuthor

  \Affil 1
    \Division        Department of Mathematics
    \Organization    Central University of Haryana
    \City            Mohindergarh
    \Country         India
  \endAffil

  \datesubmitted    May 13, 2025\enddatesubmitted
  \daterevised      August 1, 2025\enddaterevised
  \dateaccepted     August 2, 2025\enddateaccepted

  \UDclass
    ??? %\?42C15; 46B15
  \endUDclass

  \title
    Matrix-Valued Metaplectic Gabor Frames in $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$
  \endtitle

  \abstract
    The concept of matrix-valued metaplectic Gabor system is introduced in $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.
    We present that every matrix-valued metaplectic Gabor system is not a Gabor frame in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.
    Necessary and sufficient conditions for matrix-valued metaplectic and Gabor systems are obtained to make them frame %\?frames
    in $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. Also, the frame operator is represented in terms of trace %\?
    in  $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. Further, the concept of scalability for both the systems is introduced and some interesting properties of the Gabor frame and the scalable Gabor frame corresponding to a bounded linear operator~$K$
    are obtained. Some necessary and sufficient conditions for the bounded linear metaplectic Gabor atom,
    generating a frame and $K$-frame for $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$, are given.
    Finally, we show that for some particular sequences, the scalable metaplectic Gabor frame
    %\? $\{a_{(\mathtt{y}, \mathtt{\eta})}\pi_{A}(\mathtt{y}, \mathtt{\eta})g\}$
    $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$
    %\?\mathtt{y}, \mathtt{\eta} пока \roman{y} и просто  \eta
    is a $K$-metaplectic Gabor frame for $L^2(\Bbb{R}^d)$
    and to illustrate the findings, examples, and counterexamples are provided. %\?
  \endabstract

  \keywords
    frame,
    Gabor atom,
    metaplectic Gabor frame,
    Gabor frame,
    $K$-frame,
    scalable frame
  \endkeywords

\endtopmatter

\head
  1. Introduction
\endhead

	The basic idea of signal breakdown in elementary signals was first observed by Gabor [1].
Based upon this incredible idea of Gabor, Duffin and Schaeffer [2]
developed frames in the context of separable Hilbert spaces to study the nonharmonic expansion of functions.
The concept of frame presented by Duffin and Schaeffer was later examined by Young [3].
Daubechies et al. [4] revived and revitalized Hilbert frames by demonstrating their significance for data processing.
Since a frame being redundant, allows nonunique representations of signals with controlled bounds. This redundancy is helpful when there is noise, data loss, or other distortions, as it can reconstruct the signal even if some of the information is lost.
	A countable sequence $\{y_{i}\}_{i\in I}$ constitutes frame %\?a
for a Hilbert space $H$,
if the inequality $\alpha_{0}\|y\|^{2}\leq \sum_{i=1}^{\infty}|\langle y, y_{i}\rangle|^{2}\leq \beta_{0}\|y\|^2$ is satisfied
for all $y\in H$
and $0<\alpha_{0}\leq \beta_{0} < \infty$. The sequence $\{y_{i}\}_{i\in I}$  is called
a~Bessel sequence %\? a {\it Bessel sequence\/} %\?вообще нет \it
if only the upper inequality is satisfied. Also, if both the bounds coincide then $\{y_{i}\}_{i\in I}$ is a tight frame. Furthermore, $\{y_{i}\}_{i\in I}$ is
a~Parseval's frame %\?
if $\alpha_{0}= \beta_{0}=1$. For more information related to the frame, one can refer to [5--12]. The most commonly used frames in time-frequency analysis, compressed sensing, radar and sonar signal processing, communication systems, and biometric recognition; 
is the Gabor frame. %\?

	Traditionally, the concept of an orthonormal basis was used to retrieve a signal's information %\?
to receive all the information. Later, tight frames---a more adaptable system---were developed to represent any vector in terms of frame components. While it is true that every frame is not a tight frame, although some can be, particularly Parseval frames. Those frames which  can be made Parseval are called scalable frames. %\?{\it scalable frames}.
Kutyniok et al. [13] presented the idea of the scalable frames, which are the scaled pieces themselves. However, Kutyniok's common scaling approach does not hold for many frames. Moreover, the approach is not appropriate for real-world circumstances due to the relatively large number of zero coefficients. As the real-world scenario requires the contribution of each vector used in the investigation. Later on, scalable frames were changed by Casazza et al. [14], who also presented a novel concept for scaling frame components.

	Some specific extensions of frames such as $K$-frame %\?$K$-frames
[15], nonlinear frames [16], $K$-atomic [13] and approximative $K$-atomic decomposition %\?decompositions
[17], scalable frames [11], controlled frames [12], controlled $K$\=frames [10],
and continuous $Kg$-fusion  %\?$K$-$g$-fusion
frames [18] have been introduced for various theoretical and practical uses. $K$-frames are generalized ordinary frames, and even though their span limit is limited to the range of~$K$, their generality makes them practically significant.

	In fields such as geophysics, medical imaging, image processing, and data frequently takes %\?
the form of multichannel signals, where each ``sample'' is a vector or matrix instead of a scalar. Matrix-valued Gabor and Wavelet frames allow us to analyze multichannel signals coherently by treating all channels together. These systems were firstly structured by Xia and Suter [19]
and Antol\'{\i}n and Zalik [20] in $L^2(\Bbb{R}, \Bbb{C}^{l\times l}) $.
Vashisht [21] studied various properties of matrix-valued frames using a bounded linear operator in $L^2 (\Bbb{R}^d, \Bbb{C}^{s\times r})$. In 2023, Jindal et al. [22] interpret 
the frame conditions in different groups including the extended affine and the Weyl--Heisenberg groups.

	The most commonly used representation of a signal is STFT (Short Time Fourier Transform), %\?(short time Fourier transform)
as it can locate the local time-frequency behavior of a signal.
However, STFT %\?один раз используется
often suffers from the trade-off between the time-frequency.
To resolve this problem a more general
$TF$-(Time-Frequency) %\? это сокращение  не используется time-frequency и как оформить
distribution,
that is, the Wigner distribution is used. Wigner distributions were introduced by Wigner [23] in 1932.
Later, Ville [24] discovered that they had origins in signal analysis.
To provide a better environment for the signal representation, more general distributions such
as metaplectic Wigner distribution %\?distributions
are used.
Cordero and Rodino [25] introduced metaplectic Wigner distributions in 2022.
In the context of metaplectic Wigner distributions, Cordero and Giacchi [26]  introduced the
metaplectic Gabor frames (MGFs) 
%\?$(MGF)$ зачем в $$, и сокращение к мн.ч., это сокращение используется много, но то прямо, то нет
as a generalization of Gabor frames in 2024.

	Motivated by the research work of Antol\'{\i}n and Zalik [20], who introduced
matrix-valued Gabor frames (MVGFs) 
%$(MVGF)$ \?зачем в $$, и сокращение к мн.ч., это сокращение используется много, но то прямо, то нет
in $L^2(\Bbb{R}, \Bbb{C}^{l\times l})$, later studied by
Vashisht et al. [22]  %\?выше написано Jindal et al. [22] и это первый автор в статье [22]
in $L^2(\Bbb{R}, \Bbb{C}^{l\times l})$,
and the $MGF$ introduced by Cordero [27] %\? Cordero and Giacchi [27] там 2 автора
in $L^2(\Bbb{R}^d)$,
we extend the notion of $MVGF$ and $MGF$ in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.
Therefore, in this article, we obtained some interesting results by interplaying between Gabor and $MGF$ in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. We obtain various necessary and sufficient conditions for
the Gabor and metaplectic Gabor systems to be a frame for $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$.
The frame operator is represented in terms of trace in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$, and the concept of scalability for both the systems is introduced, and some new results are constructed using a~bounded linear operator $K$.  Lastly, we have shown that under a particular condition, scalable $MGF$ becomes
a $K$-$MGF$ %\?ниже  $K$-MGF
frame for $L^2(\Bbb{R}^d)$.
Examples and counterexamples are also constructed.

	The structure of the article is as follows:
	\Sec*{Section 2} includes basic terminologies and notations that will be used in the article.
\Sec*{Section 3} offers the main findings of the article. This section starts with the $MVGF$ definition. Firstly, we prove the interplay between the $MVGF$ and the corresponding Gabor frame, followed by some examples and counterexamples. In the next theorem, we prove the result corresponding to Fourier coefficients in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. Later, we provide the representation of the frame operator in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. In
\Par{Theorem 3.5}{Theorems 3.5} and \Par{Theorem 3.7}{3.7}  we obtain some interesting results of the Gabor frames. For a given example, we observe that every matrix-valued metaplectic Gabor system is not a Gabor frame in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.
In \Par*{Theorem 3.9}, the necessary and sufficient conditions are obtained for the $MGF$ in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. In \Par*{Theorem 3.11}, we discuss some properties of the $MGF$ corresponding to a bounded linear operator.
The required conditions for the matrix-valued metaplectic Gabor system to be a frame for $L^2(\Bbb{R}^d)$ are given.
We discuss the conditions for which the scalable $MGF$ is a~$K$-frame. Also, we provide some examples and counterexamples to enhance our findings.
Finally, \Sec*{Section 4} concludes the study by summarizing the obtained results and offering an overview of findings and insight of possible future work. %\?
	
	\head
2. Preliminaries
\endhead
	This section includes the basic terminologies relevant to the article. We will use the following notations:

	$\Delta_{\mu,\nu}=\aleph{\mu}\varpi_{\nu}$ is used for the Gabor system where $\aleph{\mu}$ is the modulation operator with parameter $\mu$ and $\varpi_{\nu}$ is the translation operator with parameter $\nu$.

$\daleth$ is the normalized rescaling operator. %\?может, с абзаца сделать и дальше

	$\Bbb{Z}$, $\Bbb{C}$, and~$\Bbb{N}$ denote the set %\?sets
of integers, complex, and natural numbers respectively. 

$\Bbb{R}^{d}$ is $d$-dimensional Euclidean space. 

$S(\Bbb{R}^d)$ and $S'(\Bbb{R}^d)$ represent the Schwartz function space and its topological dual respectively.

$\Cal{O}_{l\times l}$ and $\Cal{I}_{l\times l}$ denote the null and identity matrices of order $l$.

$B(\Cal{H})$ and $GL(\Bbb{R}^{2d})$ denote the spaces of all bounded linear operators over $\Cal{H}$ and the general linear group of matrices over $\Bbb{R}^{2d}$ respectively.

Throughout the article, the bold letters
$\boldkey{f}$, $\bold{g}$, $\bold{x}$, $\bold{S}$, $\bold{J}$, %\?было \textbf здесь писала \bold
$\boldkey{I}_{l}$, and $\boldkey{f}_{0}$ %\?было\textit{\textbf  здесь писала \boldkey
are used for matrix-valued functions.

$I$ denotes a countable set and $\beth$ is a subset of $\Bbb{R}^{2d}$.
	
$L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l})$ is the matrix-valued space defined as:
	$$
L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l})=
	\left\{\bold{x}() %\?()
=
\bmatrix
		x_{11}()&\cdots&x_{1l}()\\ %\?()
		x_{21}()&\cdots&x_{2l}()\\
		\vdots&\cdots&\vdots\\
		x_{l1}()&\cdots&x_{ll}()
	\endbmatrix;\
x_{\roman{ij}}  %\?\mathsf{ij}
\in L^{2}(\Bbb{R}^{d}),\
1\leq \roman{m,n} %\?\mathsf{m,n}
\leq l
\right\}.
	$$
	The space $L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l})$, corresponding to the norm
	$$
\|\bold{x}\|_2^{2}=\sum_{1\leq \roman{m,n} %\?\mathsf{m,n}
\leq l}\ \int\limits_{\Bbb{R}^{d}}|x_{\roman{ij}} %\?{\mathsf{ij}}
(y)|^2dy,\quad
\bold{x}\in L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l}),
$$ is a Banach space.
	The integral of the function $\bold{x} \in L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l})$ is defined as:
	$$
	\int\limits_{\Bbb{R}^{d}}\bold{x}(y)dy=\bmatrix
		\int\limits_{\Bbb{R}^{d}}x_{11}(y)dy&\cdots&\int\limits_{\Bbb{R}^{d}}x_{1l}(y)dy\\
		\int\limits_{\Bbb{R}^{d}}x_{21}(y)dy&\cdots& \int\limits_{\Bbb{R}^{d}}x_{2l}(y)dy\\
		\vdots&\cdots&\vdots\\
		\int\limits_{\Bbb{R}^{d}}x_{l1}(y)dy&\cdots&\int\limits_{\Bbb{R}^{d}}x_{ll}(y)dy
	\endbmatrix.
	$$
	The inner product $\langle \bold{f, g}\rangle$ for $\bold{f, g}\in L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ is defined as:
	$$
\langle \bold{f, g}\rangle = \int\limits_{\Bbb{R}^{d}}\bold{f}^*\bold{g}d\bold{x},
$$
where $\bold{f}^*$ is the transpose of $\bold{f}$.

	The translation and the modulation operators in $L^{2}(\Bbb{R}^d, \Bbb{C}^{l\times l})$ with their parameters
$\mu,\nu \in \Bbb{R}^d$ are defined as
$\aleph{\mu}\bold{f}(\bold{x})=\bold{f}(\bold{x}-\mu)$ and $\varpi_{\nu}\bold{f(x)}=e^{2\pi \iota \bold{\nu x}}\bold{f(x)}$ respectively.
	
\demo{Definition 2.1 \rm[15]}
		A sequence of vectors $\{y_{i}\}$ is said to be a $K$-frame %\?{\it $K$-frame\/}
for a Hilbert space $H$ if it satisfies
		$$
\alpha_{0}\|K^*y\|^2\leq \sum_{i\in I}|\langle y, y_{i} \rangle|^2 \leq \beta_{0}\|y\|^2
\quad\text{for all}\ y\in H,
$$
where $K$ is a bounded linear operator on the Hilbert space $H$.

		If the sequence $\{y_{i}\}$ satisfies
$$\sum_{i\in I}|\langle y, y_{i} \rangle|^2=\|K^*y\|^2,
$$ then the frame will be a Parseval $K$-frame.
	\enddemo

	\demo{Definition 2.2 \rm [26]}
		A symplectic matrix is a matrix ${\boldkey{P}}\in \Bbb{R}^{2l\times 2l}$ if for some
		$${\boldkey{J}}=\bmatrix
			\Cal{O}_{l\times l}&\Cal{I}_{l\times l}\\
			-\Cal{I}_{l\times l}&\Cal{O}_{l\times l}
		\endbmatrix,
		$$
		the equality
		${\boldkey{P}}^{{\boldkey{T}}}{\boldkey{J}}{\boldkey{P}}={\boldkey{J}}$ holds.
	\enddemo

	\demo{Definition 2.3 \rm [26]}
		A group consisting of elements of the form
		$$\bmatrix
			1&a&b\\
			0&1&c\\
			0&0&1
		\endbmatrix,
\quad\text{where }  a, b, c\in \Bbb{R},
$$
		is called the Heisenberg group. %\?{\it Heisenberg group}.
A metaplectic operator $V:L^2(\Bbb{R}^d)\longrightarrow L^2(\Bbb{R}^d)$ %\?\to еще есть
is a unitary operator satisfying $V^{-1}\theta(z;\tau)V=\theta({\boldkey{P}}z;\tau)$,
where $z=(x, \xi)\in \Bbb{R}^{2d}$, ${\boldkey{P}}$ is the symplectic matrix, and $\theta=e^{2\pi \iota \tau-\iota \xi.x}\pi(z(x, \xi))$
is the Schrodinger representation of the Heisenberg group.
	\enddemo

	\demo{Definition 2.4 \rm [26]}
		The metaplectic Wigner distribution is defined as: $W_{V}(p, q)=\widehat{V}(p\bigotimes \overline{q})$, where $p, q\in L^2(\Bbb{R}^d)$, $\widehat{V}$ is the metaplectic operator, and $p\bigotimes \overline{q}(z_{1}, z_{2})=p(z_{1})\overline{q}(z_{2})$ is the tensor product of $p$ and~$q$.
	\enddemo

	\demo{Definition 2.5 \rm [26]}
		The operator $\pi_{A}(\roman{y}, {\eta}):S(\Bbb{R}^d)\longrightarrow S'(\Bbb{R}^d)$ is called
metaplectic atom %\?артикль {\it metaplectic atom\/}
if
$$
\langle f, \pi_{A}(\roman{y}, {\eta})g \rangle=W_{A}(f, g)(\roman{y}, {\eta}), (\roman{y}, {\eta})\in \beth \subseteq \Bbb{R}^{2d}
\quad\text{for all } f, g\in S(\Bbb{R}^d).
$$
		The representation of metaplectic Gabor atom %\?
is given by Cordero [27] %\?Cordero and Giacchi [27] там 2 автора
as
		$$\pi_{A}(\roman{y}, {\eta})=\alpha_{E}e^{-2\pi \iota E_{11}{E_{12}^{-\nu}\xi.x}}\mu_{E_{22}^{-\nu}\xi}(\nu_{E_{12}E_{22}^{-1}(E_{11}-E_{12}{E_{22}}^{-1}E_{21})x})\daleth_{E_{22}E_{12}^{-1}},
$$
		where
$\alpha_{E}=\sqrt{\frac{|E|}{|E_{22}E_{12}|}}$    and
		$$
E=
		\bmatrix
			E_{11}&E_{12}\\
			E_{21}&E_{22}
		\endbmatrix,
 		\text{ and } |E|\ne 0.
		$$
	\enddemo

	\demo{Definition 2.6 \rm [26]}
		The metaplectic Gabor system $\{\pi_{A}(\roman{y}, {\eta})g\}$ is said to be a $MGF$ for $L^2(\Bbb{R}^d)$ if it satisfies
$$
\alpha_{0}\|f\|^2\leq \sum_{\roman{y}, {\eta} \in \beth} |\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2 \leq \beta_{0}\|f\|^2
$$
for all $f\in L^2(\Bbb{R}^d)$ and $\alpha_{0}, \beta_{0}>0$.
	\enddemo

	\demo{Definition 2.7 \rm[26]}
		The normalized rescaling operator for some $E\in GL(d,\Bbb{R})$, $\daleth: L^2(\Bbb{R}^d)\longrightarrow L^2(\Bbb{R}^d)$ is defined as:
$$\daleth_{E} f(.) = |E|^{\frac{1}{2}}f(E.) %\?.
\quad\text{for all } f\in L^2(\Bbb{R}^d).
$$
	\enddemo

	\demo{Definition 2.8 \rm[28]}
		The sequence $\{b_{n}\}$ is said to be seminormalized %\?{\it seminormalized\/}
if for some positive numbers $a$ and $b$, the inequality
$$a\leq b_{n}\leq b\quad\text{for all } n\in \Bbb{N}
$$
		is satisfied.
	\enddemo

\demo{Definition 2.9 \rm[29]}
		The Schwartz function space $S(\Bbb{R}^d)$ is the space of all those functions $f\in \Bbb{C}^{\infty}(\Bbb{R}^d)$ which decays rapidly such that
$$
\sup_{y\in \Bbb{R}^d}|y^c\partial^d f(y)|<\infty\quad \text{for all } c, d \in \Bbb{N}^d,
$$
		where $\Bbb{C}^{\infty}(\Bbb{R}^d)$ is the space of smooth functions.

		The topological dual of $S(\Bbb{R}^d)$ is the space $S'(\Bbb{R}^d)$ containing the tempered distributions.
	\enddemo

	\proclaim{Theorem 2.10 \rm[30]}
		For some $\tau^{d}=[0,b]^d$, $b>0$, and  $f\in L^{2}(\tau^{d})\subset L^{2}(\Bbb{R}^{d})$, the following equality holds:
		$$
\sum_{k\in \Bbb{Z}^{d}}|c_{k}|^{2}=\frac{1}{b^{d}}|f|^{2},\quad\text{where }
c_{k}=\frac{1}{b^{d}}\int\limits_{\tau^{d}}f(x)e^{-2\pi \iota k.x}dx.
$$
	\endproclaim

For further studies on metaplectic Gabor frames and related concepts one can refer %\? to
[11,\,25--27,\,29].
	
\head
3. Main Results
\endhead

	Antol\'{\i}n and Zalik [20] first introduced the concept of matrix-valued systems. Based upon this idea of matrix-valued systems, Vashisht et al. [22] %\? выше писали Jindal et al. [22]
proved that the frame conditions in $L^2(\Bbb{R})$, could be extended to $L^2({\Bbb{R}, \Bbb{C}^{l\times l}})$ if we take a system of scalar matrix-valued functions and each function constitutes frame for $L^2(\Bbb{R})$.
The conditions given by Vashisht et al. [22] %\? выше писали Jindal et al. [22]
are extended to the multidimensional matrix-valued Gabor and metaplectic Gabor systems and prove that the matrix-valued frame generating system need not be a scalar matrix. It can have different diagonal elements.
We start the section by introducing $MVGF$ in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$.

	\demo{Definition 3.1}
		A sequence of vectors $\{\Delta_{\mu,\nu}{\boldkey{g}} %\?выше g было прямое
{\boldkey{I}}_{l}\}_{\mu,\nu\in \Bbb{Z}^{d}}$ is
a $MVGF$ for $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ if it satisfies
$$
\alpha_{0}\|\bold{f}\|^2\leq \sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2} \leq \beta_{0}\|\bold{f} %\?в оригинале то прямо, то нет
\|^2\quad \text{for all } \bold{f}\in L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})
 \text{ and }
\alpha_{0},\beta_{0}>0.
$$
	\enddemo

	Following is the necessary and sufficient condition for the matrix-valued Gabor system to be the $MVGF$ in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$.

	\proclaim{Theorem 3.2}
		The matrix-valued sequence
$\{\Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}\}_{\mu,\nu\in \Bbb{Z}^{d}}=
\{\diag(\Delta_{\mu,\nu}g, \Delta_{\mu,\nu}g, \dots , \Delta_{\mu,\nu}g)\}_{\mu,\nu\in \Bbb{Z}^{d}}$ is a MVGF
in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$ iff $\{\Delta_{\mu,\nu}g\}_{\mu,\nu\in \Bbb{Z}^{d}}$
constitutes a Gabor frame in $L^{2}(\Bbb{R}^{d})$.
	\endproclaim

	\demo{Proof}
		Let $\{\Delta_{\mu,\nu}g\}_{\mu,\nu\in \Bbb{Z}^{d}}$ be a Gabor frame for $L^{2}(\Bbb{R}^{d})$ with frame bounds $\alpha_{0}$ and $\beta_{0}$.
Then, for any ${\boldkey{f}}=[f_{\roman{mn}} %\?\mathsf{mn}
]_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}$ in $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$, we have
		$$
\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}=\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \diag(\Delta_{\mu,\nu}g, \Delta_{\mu,\nu}g, \dots , \Delta_{\mu,\nu}g), [\bold{f}]\rangle \|^{2}=\sum_{\mu,\nu\in \Bbb{Z}^{d}}\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}|\langle\Delta_{\mu,\nu}g, f_{\roman{mn}} %{\mathsf{mn}}
\rangle|^{2}.
$$
		Since $\Delta_{\mu,\nu}g$ is given to be a Gabor frame with its bounds $\alpha_{0}$ and $\beta_{0}$ in $L^{2}(\Bbb{R}^{d})$;
therefore,
		$$
\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2} \geq \alpha_{0}\sum_{1
\leq \roman{m,n} %\mathsf{m, n}
\leq l}\|f_{\roman{mn}} %{\mathsf{mn}}
\|^{2}=\alpha_{0}\|{\boldkey{f}}\|^{2}.
$$
		Similarly,
$$
\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}\leq \beta_{0}\|{\boldkey{f}}\|^{2}.
$$
		Conversely, let $\{\Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}\}_{\mu,\nu\in \Bbb{Z}^{d}}$ be a $MVGF$ for $L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$ with its bounds $\alpha_{0}$ and $\beta_{0}$.
Consider an arbitrary but fixed function  $f\in L^{2}(\Bbb{R}^{d})$. Then, for
		$$
{\boldkey{f}}_{0}=
		\bmatrix
			(f,0,\dots,0) & (0,0,\dots,0) & \cdots & (0,0,\dots,0) \\
			(0,0,\dots,0) & (f,0,\dots,0) & \cdots & (0,0,\dots,0) \\
			\vdots & \vdots & \cdots & \vdots\\
			(0,0,\dots,0) & (0,0,\dots,0)&  \cdots & (f,0,\dots,0)
		\endbmatrix
		$$
		we have
		$$
\align
			\alpha_{0}l\|f\|^{2}
&=\alpha_{0}\|{\boldkey{f}}_{0}\|^{2}\leq \sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}I_{l}, {\boldkey{f}}_{0}\rangle\|^{2}=l\sum_{\mu,\nu\in \Bbb{Z}^{d}}|\langle \Delta_{\mu,\nu}g, f\rangle|^{2}\leq \beta_{0}\|{\boldkey{f}}_{0}\|^{2}=\beta_{0}l \|f\|^{2},
\\
{\alpha_{0}}{}\|f\|^{2}
&\leq\sum_{\mu,\nu\in \Bbb{Z}^{d}}|\langle \Delta_{\mu,\nu}g, f\rangle|^{2}\leq{\beta_{0}}{}\|f\|^{2}.
\qed
\endalign
$$		
	\enddemo

	From \Par*{Theorem 3.2}, we obtain that %\?
every matrix-valued Gabor system is a Gabor frame for $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ iff it is a Gabor frame in $L^2(\Bbb{R}^d)$. Furthermore, \Par*{Theorem 3.2} holds for the diagonal matrix-valued functions with nonzero values, and the upcoming example illustrates this result.

	\demo{Example 3.1}
		Consider the frame $g=\{(1,0,0,0,1), (0,1,0,0,1), (0,0,1,0,1), (0,0,0,1,1), (0,0,0,0,2)\}$ for $L^2(\Bbb{R}^5)$ and the corresponding matrix-valued function
$$
{\boldkey{g}}=\bmatrix
			(1,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1)\\
			(0,0,0,0,1) & (0,1,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1)\\
			(0,0,0,0,1) & (0,0,0,0,1) & (0,0,1,0,1) & (0,0,0,0,1) & (0,0,0,0,1)\\
			(0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,1,1) & (0,0,0,0,1)\\
			(0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,1) & (0,0,0,0,2)
		\endbmatrix;
$$
then for any diagonal matrix-valued function
${\boldkey{f}}$ in $L^2(\Bbb{R}^5, \Bbb{C}^{5\times 5})$
$$
{\boldkey{f}}=
		\bmatrix
			(f,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0)\\
			(0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) \\
			(0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0)\\
			(0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0)\\
			(0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0) & (0,0,0,0,0)
		\endbmatrix,
		$$
		then the sequence $\{\Delta_{\mu, \nu}{\boldkey{g}}{\boldkey{I}}_{l}\}$ constitutes a frame for any arbitrary but fixed translation parameter $\nu=(0,0,0,0,1)$ in $L^2(\Bbb{R}^5, \Bbb{C}^{5\times 5})$ with its frame bounds $|f|^2$.		
	\enddemo

	The following example shows that \Par*{Theorem 3.2} does not hold good %\?
if we take any diagonal entry zero.

	\demo{Example 3.2}
		Consider the functions ${\boldkey{f}}, {\boldkey{g}} \in L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ as
		$$
{\boldkey{f}}=
		\bmatrix
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0) \\
			(g,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0) \\
			\vdots & \vdots & \vdots & \vdots\\
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0)
		\endbmatrix_{l\times l}
		$$
and
$$
{\boldkey{g}}=
		\bmatrix
			(1,0,\dots,0) & (1,0,\dots,0)  &\cdots & (1,0,\dots,0) \\
			(1,0,\dots,0) & (1,1,\dots,0) &\cdots & (1,0,\dots,0)  \\
			\vdots & \vdots & \vdots & \vdots\\
			(1,0,\dots,0) & (1,0,\dots,0) &\cdots & (1,0,\dots,1)
		\endbmatrix_{l\times l},
		$$
where $g=\{(1,0,\dots,0), (1,1,\dots,0),\dots,(1,0,\dots,1)\}$ is a frame in $L^2(\Bbb{R}^d)$ and the MVGF with translation parameter $\nu=(1,0,\dots , 0)$ gives
		$$
\align
\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}
&=
		\sum_{\mu\in \Bbb{Z}^{d}}\left\|\biggglangle %\?нужно правильно \biggglangle увеличить  и дальше
\aleph{\mu}
		\bmatrix
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0)
\\
			(0,0,\dots,0) & (1,0,\dots,0) &\cdots & (0,0,\dots,0)
\\
			\vdots & \vdots & \vdots & \vdots
\\
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (1,0,\dots,0)
		\endbmatrix_{l\times l}. %\?.
\right.
\\
&
\left.
\bmatrix
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0)
\\
			(g,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0)
\\
			\vdots & \vdots & \vdots & \vdots
\\
			(0,0,\dots,0) & (0,0,\dots,0) &\cdots & (0,0,\dots,0)
		\endbmatrix_{l\times l}
\bigggrangle
\right\|^2.
\endalign
		$$
		This implies $$\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}=0.
$$
		This proves that $\{\Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}\}$ is not a frame in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$.
	\enddemo

	In the upcoming example we will observe that \Par*{Theorem 3.2} does not hold if we take a nondiagonal arbitrary matrix element in $L^2({\Bbb{R}^4, \Bbb{C}^{4\times 4}})$.

	\demo{Example 3.3}
		Consider an arbitrary matrix-valued function
$$
{\boldkey{f}}=
		\bmatrix
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0) & (0,0,0,0)\\
			(0,g,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0) \\
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0)\\
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0)
		\endbmatrix,
		$$
		and the matrix-valued frame for $L^2(\Bbb{R}^4)$ as $\{(1,0,0,1), (0,1,0,1), (0,0,1,1), (0,0,0,2)\}$ and the corresponding matrix
		$$
{\boldkey{g}}=
		\bmatrix
			(1,0,0,1) & (1,0,0,1) & (0,0,0,1) & (0,0,0,1)\\
			(0,0,0,1) & (0,1,0,1) & (0,0,0,1)& (0,0,0,1) \\
			(0,0,0,1) & (0,0,0,1) & (0,0,1,1)& (0,0,0,1)\\
			(0,0,0,1) & (0,0,0,1) & (0,0,0,1)& (0,0,0,1)
		\endbmatrix.
		$$
		Then for $L^2(\Bbb{R}^4, \Bbb{C}^{l\times l})$, with $\nu=(0,0,0,1)$ we have
		$$
\align
\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}
&=
		\sum_{\mu\in \Bbb{Z}^{d}}
\left\|
\biggglangle %\?
\aleph{\mu}
		\bmatrix
			(1,0,0,0) & (1,0,0,0) & (0,0,0,0) & (0,0,0,0)\\
			(0,0,0,0) & (0,1,0,0) & (0,0,0,0)& (0,0,0,0) \\
			(0,0,0,0) & (0,0,0,0) & (0,0,1,0)& (0,0,0,0)\\
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,1)
		\endbmatrix.
\right.
\\
&
\left.
\bmatrix
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0) & (0,0,0,0)\\
			(0,g,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0) \\
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0)\\
			(0,0,0,0) & (0,0,0,0) & (0,0,0,0)& (0,0,0,0)
		\endbmatrix
\bigggrangle %\?
\right\|^2.
\endalign
		$$
		This gives
		$$\sum_{\mu,\nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu,\nu}{\boldkey{g}}{\boldkey{I}}_{l}, {\boldkey{f}}\rangle\|^{2}=0.
$$
	\enddemo

	This proves that the result does not hold for a nondiagonal matrix-valued function.
	
	Before representing the frame operator in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ corresponding to the matrix-valued function,
	we generalizes \Par*{Theorem 2.10} in the forecoming %\?forthcoming
result, in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ to obtain the relation
between the matrix-valued function and its Fourier coefficients in $d$-dimensions.

	\proclaim{Theorem 3.3}
		For some matrix-valued function ${\boldkey{f}}\in L^{2}(\tau^{d}, \Bbb{C}^{l\times l})\subset L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$, where $\tau^d=[0, b]^d$, $b>0$, the following equality holds:
		$$
\sum_{k\in \Bbb{Z}^{d}}\|c_{k}\|^{2}=\frac{1}{b^{d}}\|{\boldkey{f}}\|^{2},\quad \text{where } c_{k}=\frac{1}{b^{d}}\int\limits_{\tau^{d}}{\boldkey{f}}(x)e^{-2\pi \iota k.x}dx. $$
	\endproclaim

	\demo{Proof}
		For ${\boldkey{f}}=[f_{\roman{mn}} %\mathsf{mn}
]_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l} \in L^{2}(\tau^{d}, \Bbb{C}^{l\times l})\subset L^{2}(\Bbb{R}^{d},\Bbb{C}^{l\times l})$ and $c_{k}=\frac{1}{b^{d}}\int\nolimits_{\tau^{d}}{\boldkey{f}}(x)e^{-2\pi \iota k.x}dx$. %\?объединить предложения
		We have
		$$
\align
			\sum_{k\in \Bbb{Z}^{d}}\|c_{k}\|^{2}=\sum_{k\in \Bbb{Z}^{d}}
\biggl\|\frac{1}{b^{d}}\int\limits_{\tau^{d}}{\boldkey{f}}(x)e^{-2\pi \iota k.x}dx
\biggl\|^{2}&=\sum_{k\in \Bbb{Z}^{d}}\biggr\|
\frac{1}{b^{d}}\int\limits_{\tau^{d}}
[f_{\roman{mn}} %\?{\mathsf{mn}}
]e^{-2\pi \iota k.x}dx\biggr\|^{2},
\\
&=
\sum_{k\in \Bbb{Z}^{d}}\biggl\|\frac{1}{b^{d}}\int\limits_{\tau^{d}}
[f_{\roman{mn}} %{\mathsf{mn}}
. %\?
e^{-2\pi \iota k.x}]dx\biggr\|^{2},
\\
			&=\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}\sum_{k\in \Bbb{Z}^{d}}\biggl|\frac{1}{b^{d}}\int\limits_{\tau^{d}}f_{\roman{mn}} %{\mathsf{mn}}
.e^{-2\pi \iota k.x}dx\biggr|^{2}.
		\endalign
$$
		Therefore, using \Par*{Theorem 2.10}, we have
		$$
			\sum_{k\in \Bbb{Z}^{d}}\|c_{k}\|^{2}
=\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}
\frac{1}{b^{d}}|f_{\roman{mn}} %\mathsf{mn}
|^{2}
=\frac{1}{b^{d}}\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}|f_{\roman{mn}} %\mathsf{mn}
|^{2}=\frac{1}{b^{d}}\|{\boldkey{f}}\|^{2}.\qed
$$
\enddemo

	In the upcoming theorem, the representation of frame operator %\?
for $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$ is given.

	\proclaim{Theorem 3.4}
		Consider the family $\{\Delta_{\mu, \nu}{\boldkey{g}}\}_{\mu, \nu\in \Bbb{Z}^{d}}$ and
${\boldkey{g}}=[g_{\roman{mn}} %{\mathsf{mn}}
]$
where $g_{\roman{mn}} %{\mathsf{mn}}
\in L^2(\Bbb{R}^d)$ such that $g_{\roman{mn}}$ %{\mathsf{mn}}
is $\Bbb{Z}^{d}$-periodic.
If the family $\{\Delta_{\mu, \nu}{\boldkey{g}}\}$ constitutes a MVGF with bounds $A$ and $B$ then the inequality
		$$
tr\langle A{\boldkey{f}}, {\boldkey{f}}\rangle\leq tr
\biglangle {\boldkey{f}},
\sum{\boldkey{g}}{\boldkey{g}}^*{\boldkey{f}}\bigrangle \leq tr\langle B{\boldkey{f}}, {\boldkey{f}}\rangle
$$
holds.
	\endproclaim

	\demo{Proof}
		Since
		$$
			S{\boldkey{f}}=\sum_{\mu,\nu\in \Bbb{Z}^d}\langle {\boldkey{f}}, \Delta_{\mu, \nu}{\boldkey{g}} \rangle \Delta_{\mu, \nu}{\boldkey{g}};
\tag3.1		
		$$
		therefore, using \Tag(3.1) we have
		$$
\align
			tr\langle {\boldkey{f}}, S{\boldkey{f}}\rangle
&=tr\bigglangle {\boldkey{f}}, \sum_{\mu,\nu\in \Bbb{Z}^d}\langle {\boldkey{f}}, \Delta_{\mu, \nu}{\boldkey{g}} \rangle \Delta_{\mu, \nu}{\boldkey{g}} \biggrangle=tr\sum_{\mu,\nu\in \Bbb{Z}^d}|\langle {\boldkey{f}}, \Delta_{\mu, \nu}{\boldkey{g}} \rangle|^2.
\endalign
$$
		So
		$$
\align
			tr\langle {\boldkey{f}}, S{\boldkey{f}}\rangle = \sum_{\mu, \nu\in \Bbb{Z}^{d}}\|\langle \Delta_{\mu, \nu}{\boldkey{g}}, {\boldkey{f}}\rangle\|^{2}&=\sum_{\mu, \nu\in \Bbb{Z}^{d}}\ \biggl\|\ \int\limits_{\tau^{d}}e^{-2\pi \iota \mu.x}{{\boldkey{g}}^*(x-\nu)}{\boldkey{f}}(x)dx\biggr\|^{2},
\\
			&=\sum_{\mu, \nu\in \Bbb{Z}^{d}}\ \biggl\|\ \int\limits_{\tau^{d}}e^{-2\pi \iota \mu.x}{{\boldkey{g}}^*(x)}{\boldkey{f}}(x)dx\biggr\|^{2}.
		\endalign
$$
		Using \Par*{Theorem 3.3}, we have
		$$
			tr\langle {\boldkey{f}}, S{\boldkey{f}}\rangle
=\sum_{\mu,\nu\in \Bbb{Z}^{d}}\ \biggl\|\ \int\limits_{\tau^{d}}e^{-2\pi \iota \mu.x}{{\boldkey{g}}^*(x)}{\boldkey{f}}(x)dx\biggr\|^{2}=\sum_{\nu\in \Bbb{Z}^{d}}\|{\boldkey{g}}^*{\boldkey{f}}\|^{2}.
$$
So
$$
			tr\langle {\boldkey{f}}, S{\boldkey{f}}\rangle =\sum_{\nu\in \Bbb{Z}^{d}}\|{\boldkey{g}}^*{\boldkey{f}}\|^{2}=\sum_{\nu\in \Bbb{Z}^{d}}tr\langle {\boldkey{g}}^*{\boldkey{f}}, {\boldkey{g}}^*{\boldkey{f}}\rangle = tr\bigglangle {\boldkey{f}}, \sum_{\nu\in \Bbb{Z}^{d}}{\boldkey{g}}{\boldkey{g}}^*{\boldkey{f}}\biggrangle.
$$
		Thus, %we have
$$
S{\boldkey{f}}=\sum_{\nu\in \Bbb{Z}^{d}}{\boldkey{g}}{\boldkey{g}}^*{\boldkey{f}}.\qed
$$
	\enddemo

	In the upcoming theorems, the Gabor frame is characterized in $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$ using a bounded linear operator $K$. The scalability of Gabor frames is also discussed.

	\proclaim{Theorem 3.5}
		For an invertible operator $K:L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})\longrightarrow L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$,
$K\in B(\Cal{H})$ satisfying
$\langle K{\boldkey{f}}, K{\boldkey{g}}\rangle =\langle {\boldkey{f}}, {\boldkey{g}}\rangle $
for all ${\boldkey{f}}, {\boldkey{g}}\in L^2(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$,
the sequence $\{\Delta_{\mu, \nu}{\boldkey{g}}\}_{\mu, \nu\in \Bbb{Z}^{d}}$
constitutes a Gabor frame in $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$
iff $\{K\Delta_{\mu, \nu}{\boldkey{g}}\}$ is a Gabor frame.
	\endproclaim

	\demo{Proof}
		Suppose $A$ and $B$ are the frame bounds of the frame $\{\Delta_{\mu, \nu}{\boldkey{g}}\}$ in $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$.
Then, for some ${\boldkey{f}}\in L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$, consider
		$$
			\sum_{\mu, \nu \in \Bbb{Z}^{d}}\|\langle K\Delta_{\mu, \nu}{\boldkey{g}},{\boldkey{f}}\rangle \|^{2}=\sum_{\mu, \nu \in Z^{d}}\|\langle \Delta_{\mu, \nu}{\boldkey{g}}, K^*{\boldkey{f}}\rangle \|^{2}\leq B\|K^*{\boldkey{f}}\|^2\leq B
			\|K^*\|^2\|{\boldkey{f}}\|^2.
	    $$
		Similarly,
		$$
\align
			\|{\boldkey{f}}\|^2&=\|KK^{-1}{\boldkey{f}}\|^2
			\leq \frac{\|K\|^2}{A}\sum_{\mu, \nu\in \Bbb{Z}^d}\|\langle \Delta_{\mu, \nu}{\boldkey{g}},
K^{-1}{\boldkey{f}}\rangle\|^2, %\?,
\\
&=\frac{\|K\|^2}{A}\sum_{\mu, \nu\in \Bbb{Z}^d}\|\langle K\Delta_{\mu, \nu}{\boldkey{g}}, {\boldkey{f}}\rangle \|^2.
		\endalign $$
		This proves %\?that
$\{K\Delta_{\mu, \nu}{\boldkey{g}}\}$ is a Gabor frame with its bounds $\frac{A}{\|K\|^2}$ and $B\|K^*\|^2$.
		Converse, can be easily proved if $K=I$. %\?
	\qed\enddemo

	\demo{Definition 3.6}
		A Gabor frame $\{\Delta_{\mu, \nu}g\}$ is called
a scalable Gabor frame %\?a {\it scalable Gabor frame\/}
for a Hilbert space $H$ if for some sequence of reals $\{a_{\mu, \nu}\}$, the family $\{a_{\mu, \nu}\Delta_{\mu, \nu}g\}$ satisfies
$$
\sum_{\mu,\nu \in \Bbb{Z}^d}|\langle f, a_{\mu,\nu}\Delta_{\mu, \nu}g\rangle|^2 = \|f\|^2\quad\text{for all } f\in H.
$$
	\enddemo

	\proclaim{Theorem 3.7}
		If $\{\Delta_{\mu, \nu}g: g\in L^2(\Bbb{R})\}$ is a scalable frame corresponding to the sequence $\{a_{\mu, \nu}\}$ for $L^2(\Bbb{R})$. Then, %\?then
for some $K\in B(H)$ which preserves norm such that $K^*$ is bounded below.
Then, %\?
$\{Ka_{\mu, \nu}\Delta_{\mu, \nu}g\}$ will be a $K$-frame.		
	\endproclaim

	\demo{Proof}
		For some $h\in L^2(\Bbb{R})$, consider
		$$
			\sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2=\sum_{\mu, \nu \in \Bbb{Z}^d}|\langle K^*h, a_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2.
\tag3.2			
		$$
		But the sequence $\{\Delta_{\mu, \nu}g\}$ is scalable, so $\{a_{\mu, \nu}\Delta_{\mu, \nu}g\}$ is a Parseval frame, therefore from \Tag(3.2) we have
		$$\sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2=\|K^*h\|^2\leq \|K\|^2\|h\|^2.
$$
		Since $K$ is bounded below, so $a\|h\|^{2}\leq\|K^*h\|^2$ for some $a>0$.

		Therefore, we have
		$$ a\|h\|^{2}\leq\|K^*h\|^2=\sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2.
$$
		This gives
		$$a\|h\|^{2} \leq \sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2.
$$
		Since $K$ preserves isometry, so using $\|Kh\|^2=\|h\|^2$, we get
		$$
		a\|Kh\|^{2}\leq \sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2.
$$
		Also $\|K^*h\|^2=\|Kh\|^2$, therefore, we get
		$$a\|K^*h\|^{2}\leq \sum_{\mu, \nu \in \Bbb{Z}^d}|\langle h, Ka_{\mu, \nu}\Delta_{\mu, \nu}g\rangle |^2 \leq \|K\|^{2}\|h\|^{2}.
$$
		This proves that $\{Ka_{\mu, \nu}\Delta_{\mu, \nu}g\}$ is $K$ frame. %\?a $K$-frame.
	\qed\enddemo
	
	Cordero [26] %\?выше написано Cordero and Giacchi [26] там 2 автора
generalizes the Gabor frames to MGF.
These frames are generated by metaplectic Gabor atom %\?
defined in [27].
Inspired from the work of Cordero [26,\,27]
%\?выше написано Cordero and Giacchi [26] там 2 автора, в [27] эти же авторы Cordero and Giacchi
and L.K. %\?выше без инициалов
Vashisht [21,\,22],  %\?в [22] несколько авторов %\? выше писали Jindal et al. [22]
here we have defined the matrix-valued MGFs and observed that every matrix-valued metaplectic Gabor system is not a frame for $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.  First, we introduce the matrix-valued MGFs.
	
\demo{Definition 3.8}
		A sequence of vectors $\{\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}{\boldkey{I}}_{l}\}$ is said to
be a matrix-valued $MGF$ %\?\it
in $L^2(\Bbb{R}^d, \Bbb{C}^{l\times l})$ if it satisfies
		$$
\alpha_{0}\|\bold{f}\|^2\leq \sum_{\roman{y}, {\eta}\in \beth}\|\langle \bold{f}, \pi_{A}(\roman{y}, {\eta})\bold{g}{\boldkey{I}}_{l}\|^{2}\leq \beta_{0}\|\bold{f}\|^2\quad\text{for all } \bold{f}\in L^2(\Bbb{R}^d, \Bbb{C}^{l\times l}).
$$
	\enddemo

	Following %\?
is the example which shows that the matrix-valued metaplectic system is not a frame for $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$.

	\demo{Example 3.4}
The metaplectic Gabor atom is defined as:
$$
\pi_{A}(\roman{y}, {\eta})=\alpha_{E}e^{-2\pi \iota E_{11}{E_{12}^{-T}{\eta}.\roman{y}}}\mu_{E_{22}^{-T}{\eta}}(\nu_{E_{12}E_{22}^{-1}(E_{11}-E_{12}{E_{22}}^{-1}E_{21})\roman{y}})\daleth_{E_{22}E_{12}^{-1}} , \quad\roman{y}, {\eta} \in \beth,
$$
		where $\alpha_{E}=\sqrt{\frac{|E|}{|E_{22}E_{12}|}}$ and
		$$
E=
		\bmatrix
			E_{11}&E_{12}\\
			E_{21}&E_{22}
		\endbmatrix.
		$$
		First, we construct a metaplectic atom with
$$
E=
		\bmatrix
			2&1\\
			0&1
		\endbmatrix,
		$$
		where
		$$
		    E_{11}=[2],\qquad
			E_{12}=[1],\qquad
			E_{21}=[0],\qquad
			E_{22}=[1].
		$$
		Here, $|E|=2$, $|E_{22}E_{12}|=1$. So, $\alpha_{E}=\sqrt{2}$.

		We observe that $\nu_{E_{12}E_{22}^{-1}(E_{11}-E_{12}{E_{22}}^{-1}E_{21})\roman{y}}=\nu_{E_{12}E_{22}^{-1}E_{11}\roman{y}}=\nu_{2\roman{y}}$ which gives $$\daleth_{E_{22}E_{12}^{-1}g}=\sqrt{|E|}g(E)=\sqrt{2}g\bmatrix
			2&1\\
			0&1
		\endbmatrix.
$$
		Substituting all %\?
$$
\pi_{A}(\roman{y}, {\eta})g=\sqrt{2}e^{-2\pi \iota \times 2 \times 1 {\eta}. %\?
\roman{y}}\mu_{{\eta}}\nu_{2\roman{y}}g
\bmatrix
			2&1\\
			0&1
\endbmatrix.
$$
		Since $E\in GL(\Bbb{R}^{2d})$, we have $d=1$, so $(\roman{y}, {\eta})\in \Bbb{R}\times \Bbb{R}$.

		Let $\roman{y}=\frac{1}{2}$ and ${\eta}=1$.
Then for some translation invariant function $g$, we have
$$
\pi_{A}\Bigl(\frac{1}{2}, 1\Bigr)g=\sqrt{2}e^{-2\pi \iota}\mu_{1}g
\bmatrix
			1&0\\
			-1&0
		\endbmatrix.
$$

		Now, for some $f=\bmatrix
			0&0\\
			0&1
		\endbmatrix$,	
		$$
\align
			\langle f, \pi_{A}(\roman{y}, {\eta})g \rangle
&=\bigglangle \bmatrix
				0&0\\
				0&1
			\endbmatrix, \sqrt{2}e^{-2\pi \iota}e^{2\pi \iota}g
\bmatrix
				1&0\\
				-1&0
			\endbmatrix
			\biggrangle=\sqrt{2}\bigglangle
\bmatrix
				0&0\\
				0&1
			\endbmatrix,
\bmatrix
				g&0\\
				-g&0
			\endbmatrix
			\biggrangle,\\
			&=\sqrt{2}tr\biggl\{\bmatrix
				0&0\\
				0&1
			\endbmatrix\bmatrix
				g&-g\\
				0&0
			\endbmatrix\biggr\}=\sqrt{2}tr
\bmatrix
				0&0\\
				0&0
			\endbmatrix=0.
		\endalign
$$
		This implies $\sum|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^{2}=0$,
which means this cannot be a frame. %\?2 this и где-то that вставить
	\enddemo

	The preceding analysis demonstrates that not all matrix-valued metaplectic atoms are capable of generating a matrix-valued MGF in $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. This holds for the diagonal matrix-valued functions with nonzero values. Next, we have given a necessary and sufficient condition for matrix-valued MGF. %\?

	\proclaim{Theorem 3.9}
		The sequence of functions $\{\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}{\boldkey{I}}_{l}\}=\{\diag(\pi_{A}(\roman{y}, {\eta})g, \pi_{A}(\roman{y}, {\eta})g, \dots, \pi_{A}(\roman{y}, {\eta})g) \}$ is a frame for $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$ iff $\{\pi_{A}(\roman{y}, {\eta})g\}$ is a MGF for $L^{2}(\Bbb{R}^{d})$.
	\endproclaim

	\demo{Proof}
		Let $\{\pi_{A}(\roman{y}, {\eta})g\}$ be a $MGF$ with $A$ and $B$ as its frame bounds in $L^{2}(\Bbb{R}^{d})$.
		Consider
		$$
\align
			\sum_{\roman{y},{\eta} \in \beth}\|\langle {\boldkey{f}}, \pi_{A}(\roman{y}, {\eta})\bold{g}{\boldkey{I}}_{l}\rangle\|^{2}
&=\sum_{\roman{y},{\eta} \in \beth}\|
\langle [f_{\roman{mn}} %\mathsf{mn}
], \diag{\pi_{A}(\roman{y}, {\eta})g, \dots , \pi_{A}(\roman{y}, {\eta})g}\rangle\|^{2}
\\
&=\sum_{\roman{y},{\eta} \in \beth}\sum_{1\le \roman{m,n} %\mathsf{m, n}
\le l}|\langle f_{\roman{mn}}, %\mathsf{mn},
\pi_{A}(\roman{y}, {\eta})g \rangle|^{2}
			\leq B\sum_{1\le {\roman{m,n}} %\mathsf{m, n}
\le l}|f_{\roman{mn}} %\mathsf{mn}
|^{2}=B\|{\boldkey{f}}\|^{2}.
		\endalign
$$
		Similarly,
		$$\sum_{\roman{y},{\eta} \in \beth}\|\langle {\boldkey{f}}, \pi_{A}(\roman{y}, {\eta})\bold{g}{\boldkey{I}}_{l}\rangle\|^{2}\geq A\|{\boldkey{f}}\|^{2}.
$$
		Therefore,
		$$A\|\bold{f}\|^{2}\leq \sum_{\roman{y},{\eta} \in \beth}|\langle \bold{f}, \pi_{A}(\roman{y}, {\eta})\bold{g}\bold{I}_{l} \rangle|^{2}\leq B\|\bold{f}\|^{2}.
$$
		Converse one can easily prove. %\?
	\qed\enddemo
	
	\demo{Definition 3.10}
		The $K$-$MGF$ %\? как оформить
in $L^{2}(\Bbb{R}^{d})$ is a family of elements $\{\pi_{A}(\roman{y}, {\eta})g\}$ that satisfies
		$$\alpha_{0}\|K^*f\|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g \rangle|^2 \leq \beta_{0}\|f\|^2\quad\text{for all } f\in L^{2}(\Bbb{R}^{d}),\ \alpha_{0}, \beta_{0}>0.
$$
	\enddemo

	Next, we discuss various properties of $MGFs$ %\?$MGF$s всюду
in terms of bounded linear operators.

	\proclaim{Theorem 3.11}
		Let $\{\pi_{A}(\roman{y}, {\eta})g\}$ be a $MGF$ in $L^{2}(\Bbb{R}^{d})$ and
let $K$ be a bounded linear operator on $L^{2}(\Bbb{R}^{d})$ that preserves isometry. Then, $\{K\pi_{A}(\roman{y}, {\eta})g\}$ is a
$K$-MGF %\?%\? как оформить
for $L^{2}(\Bbb{R}^{d})$.
	\endproclaim

	\demo{Proof}
		Let $\{\pi_{A}(\roman{y}, {\eta})g\}$ be $MGF$ with its frame bounds $A$ and $B$ in $L^{2}(\Bbb{R}^{d})$. Then, for some $f\in L^{2}(\Bbb{R}^{d})$, consider
		$$
			\sum_{\roman{y}, {\eta} \in \beth}|\langle f, K\pi_{A}(\roman{y}, {\eta})g\rangle|^{2}=\sum_{\roman{y}, {\eta} \in \beth}|\langle K^*f, \pi_{A}(\roman{y}, {\eta})g\rangle|^{2}\geq A\|K^*f\|^{2}.
\tag3.3			
		$$
		Similarly,
		$$
			\sum_{\roman{y}, {\eta} \in \beth}|\langle f, K\pi_{A}(\roman{y}, {\eta})g\rangle|^{2}\leq  B\|K^*\|^{2}\|f\|^{2}.
\tag3.4			
		$$
		From \Tag(3.3) and \Tag(3.4), we obtain
		$$A\|K^*f\|^{2}\leq\sum_{\roman{y}, {\eta} \in \beth}|\langle f, K\pi_{A}(\roman{y}, {\eta})g\rangle|^{2}\leq B\|K^*\|^{2}\|f\|^{2}.
$$
		Therefore, $\{K\pi_{A}(\roman{y}, {\eta})g\}$ is a $K$-MGF for $L^{2}(\Bbb{R}^{d})$ with its frame bounds $A$ and $B\|K^*\|^2$.
	\qed\enddemo

	\proclaim{Theorem 3.12}
		The matrix-valued metaplectic Gabor system
$$\{K\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}{\boldkey{I}}_{l}\}=\{\diag(K\pi_{A}(\roman{y}, {\eta})g, K\pi_{A}(\roman{y}, {\eta})g, \dots , K\pi_{A}(\roman{y}, {\eta})g)\},
$$
where $K$ preserves isometry, is a Gabor frame for $L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$ with its bound $A$ and $\|BK^*\|^2$ iff $\{\pi_{A}(\roman{y}, {\eta})g %\?_\mathbf{}
\}$ is a $MGF$ for $L^{2}(\Bbb{R}^d)$.
	\endproclaim

	\demo{Proof}
		Let $\{\pi_{A}(\roman{y}, {\eta})g %\?_\mathbf{}
\}$ be a $MGF$	in $L^{2}(\Bbb{R}^d)$ with its frame bounds $A$ and $B$.
Then, for some ${\boldkey{f}}\in L^{2}(\Bbb{R}^{d}, \Bbb{C}^{l\times l})$, consider
		$$
\aligned
			\sum_{\roman{y}, {\eta} \in \beth}\|\langle {\boldkey{f}}, K\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}\rangle\|^{2}
&=\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}\sum_{\roman{y}, {\eta} \in \beth}|\langle f_{\roman{mn}}, %\mathsf{mn},
K\pi_{A}(\roman{y}, {\eta})g %_\mathbf{}
\rangle|^{2}
\\
&=\sum_{1\leq {\roman{m,n}} %\mathsf{m, n}
\leq l}\sum_{\roman{y}, {\eta} \in \beth}|\langle K^*f_{\roman{mn}}, %{\mathsf{mn}},
\pi_{A}(\roman{y}, {\eta})g %_\mathbf{}
\rangle|^{2}.
\endaligned
\tag3.5
		$$
		Therefore, using \Tag(3.5) we have
		$$
			A\sum_{1\leq\roman{m,n} %\mathsf{m,n}
\leq l}|K^*f_{\roman{mn}} %{\mathsf{mn}}
|\leq\sum_{\roman{y}, {\eta} \in \beth}\|\langle {\boldkey{f}}, K\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}\rangle\|^{2}\leq B\sum_{1\leq \roman{m,n} %\mathsf{m, n}
\leq l}|K^*f_{\roman{mn}} %{\mathsf{mn}}
|^{2}=B\|K^*f\|^2.
\tag3.6
		$$
		This gives
		$$
A\|K^*{\boldkey{f}}\|^2\leq\sum_{\roman{y}, {\eta} \in \beth}\|\langle {\boldkey{f}}, K\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}\rangle\|^{2}=B\|K^*f\|^2.
$$
		But $K$ preserves isometry, therefore, we have
		$$A\|{\boldkey{f}}\|^2\leq\sum_{\roman{y}, {\eta} \in \beth}\|\langle {\boldkey{f}}, K\pi_{A}(\roman{y}, {\eta}){\boldkey{g}}\rangle\|^{2}=B\|K^*\bold{f}\|^2.
$$
		Converse one can easily prove. %\?
	\qed\enddemo

	In the context of Hilbert spaces, it has been determined that no frame can be universally categorized as either a scalable frame or
$K$-frame. This concept is illustrated in the following examples. Additionally, the study presents the necessary condition for a scalable $MGF$ to be considered as a $K$-$MGF$.  %\?
First, we introduce the concept of scalable $MGF$.

\demo{Definition 3.13}
		The $MGF \{\pi_{A}(\roman{y}, {\eta} )g\}$ for $H$ is said to be scalable %\?{\it scalable\/}
if for some sequence of reals $\{a_{(\roman{y}, {\eta})}\}$, the $MGF$ generated by the sequence $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta} )g\}$ satisfies
$$
\sum_{\roman{y}, {\eta} \in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta} )g\rangle|^2=\|f\|^2\quad\text{for all } f\in H.
$$		
	\enddemo

	The forecoming %\?forthcoming
example shows that for every sequence $\{a_{\roman{y}, {\eta}}\}$ the scalable frame $\{\pi_{A}(\roman{y}, {\eta})g\}$ is not a $K$-MGF.

	\demo{Example 3.5}
		Let $\{\pi_{A}(\roman{y}, {\eta})g\}$ be a $K$-$MGF$ %\?
for $L^2(\Bbb{R}^d)$ with its frame bounds $\alpha_{0}$ and $\beta_{0}$. Then, the scalable $MGF~\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is not a $K$-frame corresponding to the sequence
		$$
 \{a_{(\roman{y}, {\eta})}\}=
 \cases
			\|\roman{y}+{\eta}\|,& \text{if } \roman{y}=(n',0,\dots,0),
 \text{ and }  {\eta}=(0,n',\dots %\?,
 0), \text{ and } (n'=1, 2, \dots %\?, зачем в скобках и предыдущее and убрать
 \infty),\\
			0& \text{otherwise}.
		\endcases
		$$
	\enddemo

	Here, in this example,
$a_{(\roman{y}, {\eta})}
=\|\roman{y}+{\eta}\|
=\|(n',n',0,\dots,0)\|=\sqrt{n^{\prime 2}+n^{\prime 2}}=\sqrt{2}n'$.
It is given that
	$$
		\alpha_{0}\|K^*f\|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2 \leq \beta_{0}\|f\|^2\quad \text{for all } f\in L^2(\Bbb{R}^d).
\tag3.7
	$$
	So, we have
$$
\aligned
		\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\rangle|^2
&=\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \sqrt{2}n'\pi_{A}(\roman{y}, {\eta})g\rangle|^2
\\
&=2n^{\prime 2}\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2,\quad\text{where } n'\in \Bbb{N}.
\endaligned
\tag3.8
	$$
	Thus, from \Tag(3.7) and \Tag(3.8), we get
	$$
		2n^{\prime 2}\alpha_{0}\|K^*f\|^2\leq\sum_{(\roman{y}, {\eta})}
|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq 2n^{\prime 2}\beta_{0}\|f\|^2\leq 2(n'+1)^2B\|f\|^2.
\tag3.9
	$$
	As $n' \longrightarrow \infty$ %\?$n'\to\infty$ 
in \Tag(3.9), we observe that $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is not
a $K$-MGF. %\? как оформить

	The above discussion leads to the following result that for certain sequences $\{a_{\roman{y}, {\eta}}\}$, the scalable $MGF \{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ constitutes a $K$-frame.

	\proclaim{Theorem 3.14}
		If $\{\pi_{A}(\roman{y}, {\eta})g\}$ is a $K$-MGF in $L^2(\Bbb R^d)$ with $\alpha_{0}$ and $\beta_{0}$  its bounds then for
a~seminormalized sequence $\{a_{(\roman{y}, {\eta})}\}$, the scalable frame generated by the sequence $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is a $K$-frame.
	\endproclaim

	\demo{Proof}
		Since $\{\pi_{A}(\roman{y}, {\eta})g\}$ is a $K$-MGF with its frame bounds
$\alpha_{0}$ and $\beta_{0}$, we have
		$$
			\alpha_{0}\|K^*f\|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2 \leq \beta_{0}\|f\|^2\quad \text{for all } f\in L^2(\Bbb R^d).
\tag3.10
		$$
		Consider
$$
			\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g \rangle|^2=\sum_{\roman{y}, {\eta}\in \beth}|a_{(\roman{y}, {\eta})}|^2|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2.
\tag3.11
$$
		But the sequence $\{a_{(\roman{y}, {\eta})}\}$ is seminormalized, therefore
according to \Par*{Definition 2.8},  %\?
there exist constants $a, b>0$ such that $a\leq |a_{(\roman{y}, {\eta})}|^2\leq b$ for all $\roman{y}$ and ${\eta}$.

		So, from \Tag(3.11), we have
		$$
a\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g \rangle|^2\leq b\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2.
$$
		Now using \Tag(3.10) and \Tag(3.11), we get
		$$a\alpha_{0}\|K^*f\|^2\leq a\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g \rangle|^2\leq b\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq b\beta_{0}\|f\|^2.
$$
		This gives
		$$a\alpha_{0}\|K^*f\|^2\leq\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g \rangle|^2\leq b\beta_{0}\|f\|^2.
$$
		Therefore, we have
		$$ A'\|K^*f\|^2\leq\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g \rangle|^2\leq B'\|f\|^2,\quad\text{where }  A'=a\alpha_{0} \text{ and }  B'=b\beta_{0}.
\tag3.12			
		$$
		Also, if $A'=B'$ in \Tag(3.12) then the sequence $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is a tight $K$-MGF. It will be Parseval $K$-MGF %\?
if $K=I$.
	\qed\enddemo

	The following example illustrates that if we take a seminormalized sequence $\{a_{\roman{y}, {\eta}}\}$, then \Par*{Theorem~3.14} holds.

	\demo{Example 3.6}
		Consider the seminormalized  sequence
$$\{a_{\roman{y}, {\eta}}\}=
		\cases
			\|\roman{y} %\mathsf{y}
+{\eta} %\mathsf{\eta}
\|,
& \text{if } \roman{y}=(1,0,\dots %\?,
0), \text{ and } {\eta}=(1,0,\dots,0),\\
			1& \text{otherwise}.
		\endcases
		$$
		and the $K$-$MGF \{\pi_{A}(\roman{y}, {\eta})g: g\in L^2(\Bbb{R}^{d})\}$, with its bounds
$\alpha_{0}$ and $\beta_{0}$. Then, the sequence
$\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$
generates a $K$-MGF for $L^2(\Bbb{R}^d)$.
	\enddemo

	Since  %\?объединить предложения
$$
	1\leq \{a_{(\roman{y}, {\eta})}\}\leq 2\quad \text{for all }
\roman{y}, {\eta}.
%\mathsf{y}, \mathsf{\eta}.
$$
	
	It is given that
	$$
		\alpha_{0}\|K^*f\|^2\leq \sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2 \leq \beta_{0}\|f\|^2\quad \text{for all } f\in L^2(\Bbb{R}^d).
\tag3.13
	$$
	Consider
$$
\sum_{\roman{y}, {\eta}\in \beth}
|\langle f, a_{\roman{y}, {\eta}}\pi_{A}
(\roman{y}, {\eta})g\rangle|^2
=
\sum_{\roman{y}, {\eta}\in \beth}
|a_{\roman{y}, %\mathsf{y},
{\eta} %\mathsf{\eta}
}|^2|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2.
\tag3.14 		
$$
	Now using equation \Tag(3.14), we have
$$
\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{\roman{y}, {\eta}}\pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq4\sum_{\roman{y}, {\eta}\in \beth}|\langle f, \pi_{A}(\roman{y}, {\eta})g\rangle|^2\quad \text{for all } f\in L^2(\Bbb{R}^d).
$$
	Therefore, using equation \Tag(3.13)
$$
\alpha_{0}\|K^*f\|^2\leq\sum_{\roman{y}, {\eta}\in \beth}|\langle f, a_{\roman{y}, {\eta}}\pi_{A}(\roman{y}, {\eta})g\rangle|^2\leq4\beta_{0}\|f\|^2.
$$
	This implies that $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is
$K$-MGF. %\?a
	
\head
4. Conclusion
\endhead
	In this article, we studied Gabor and metaplectic Gabor systems in $L^2(\Bbb{R}^d,\Bbb{C}^{l\times l})$. Inspired from the recent research work in the field of $MVGF$ in $L^2(\Bbb{R},\Bbb{C}^{l\times l})$, $MGF$, scalable frames, we observe that for the real life problems it would be interesting to observe things in multiple dimensions. Based upon this idea, firstly we define the $MVGF$ in $L^2(\Bbb{R}^d,\Bbb{C}^{l\times l})$. The necessary and sufficient conditions for the matrix-valued Gabor systems to be a frame $L^2(\Bbb{R}^d,\Bbb{C}^{l\times l})$ are obtained. Later on, the frame operator in $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$ is defined. Metaplectic Gabor atoms are defined to introduce
the $MGFs$, which are a natural extension of Gabor frames [27]. But we observe that most of the problems in signal analysis works on matrix valued functions as an input or output.
Therefore, these $MGFs$ %\?$MGF$s всюду посмотреть
are generalized as matrix-valued $MGFs$
in $L^2(\Bbb{R}^d,\Bbb{C}^{l\times l})$. We proved that not every matrix-valued metaplectic Gabor atom generates a frame for $L^2(\Bbb{R}^{d},\Bbb{C}^{l\times l})$. Also, the necessary and sufficient conditions are proved for the matrix-valued metaplectic Gabor system to be a frame. We also discussed some properties of Gabor
and $MGFs$ under the action of some bounded linear operator $K$. The necessary and sufficient conditions are proved for the scalable Gabor frame to be $K$-frame. The importance of scalable frames is very well known %\?
in the fields such as image and signal analysis, where stable reconstruction of signals is required. Here, we introduce scalable $MGFs$
and noticed %\?noted
that the $K$-scalable $MGFs$ are not always a $K$-frame unless for some particular type of sequences. Lastly, the conditions for the sequence $\{a_{(\roman{y}, {\eta})}\}$ is obtained such that $\{a_{(\roman{y}, {\eta})}\pi_{A}(\roman{y}, {\eta})g\}$ is $K$-MGF for $L^2(\Bbb{R}^d)$. Some examples and counterexamples are provided to demonstrate the outcomes.

\Refs

\ref\no 1
\by                Gabor~D.
\paper             Theory of communication. Part 1: The analysis of information
\jour              Proc. Inst. Electr. Engineers
\yr                1946
\vol               93
%\issue            -
\pages             429--457
\endref
 % Journal of the Institution of Electrical Engineers-part III: radio and communication engineering, 93(26), 429-441, (1946).

\ref\no 2
\by                Duffin~R.J. and Schaeffer~A.C.
\paper             A class of nonharmonic Fourier series
\jour              Trans. Amer. Math. Soc.
\yr                1952
\vol               72
%\issue            -
\pages             341--366
\endref

\ref\no 3
\by               Young~R.M.
      \book       An Introduction to Nonharmonic Fourier Series
      \publ       Academic
      \publaddr   New York and London
            \yr   1980
\finalinfo        Pure Appl. Math., 93
\endref

\ref\no 4
\by               Daubechies~I., Grossmann~A.,  and Meyer~Y.
\paper            Painless nonorthogonal expansions
\jour             J.~Math. Phys.
\yr               1986
\vol              27
\issue            5
\pages            1271--1283
\endref

\ref\no 5
\by               Casazza~P.G. and  Kutyniok~G. \eds
      \book       Finite Frames: Theory and Applications
      \publ       Birkh\"auser/Springer
      \publaddr   New York
            \yr   2013
\finalinfo        Appl. Numer. Harmon. Anal.
\endref

\ref\no 6
\by               Christensen~O.
      \book       Frames and Bases: An Introductory Course
      \publ       Birkh\"auser %Boston
      \publaddr   Boston
            \yr   2008
\finalinfo        Appl. Numer. Harmon. Anal.
\endref

\ref\no 7
\by               Okoudjou~K.A.
\book             Finite Frame Theory: A Complete Introduction to Overcompleteness
%\bookinfo         Proceedings of the AMS Short Course held in San Antonio, TX, January 8--9, 2015
\publaddr         Providence
\publ             Amer. Math. Soc.
\yr               2016
\finalinfo        Proc. Sympos. Appl. Math., 73
\endref

\ref\no 8
\by               Poumai~K.T. and Jahan~S.
\paper            On $K$-atomic decompositions in Banach spaces
\jour             Electron. J. Math. Anal. Appl.
\yr               2018
\vol              6
\issue            1
\pages            183--197
\endref

\ref\no 9
\by               Poumai~K.T. and Jahan~S.
\paper            Atomic systems for operators
\jour             J.~Wavelets Multiresolut. Inf. Process.
\yr               2019
\vol              17
\issue            1
\num              1850066
\size             18
\endref

\ref\no 10
\by               Nouri~M., Rahimi~A., and Najafzadeh~Sh.
\paper            Controlled $K$-frames in Hilbert spaces
\jour             J.~Ramanujan Math. Soc.
\yr               2015
\vol              4
\issue            2
\pages            39--50
\endref

\ref\no 11
\by               Rahimi~A. and Moayyadzadeh~S.
\paper            Scalable fusion frames
\jour             Iran. J. Sci.
\yr               2025
\vol              49
\issue            3
\pages            825--831
\endref

\ref\no 12
\by               Rashidi-Kouchi~M. and Rahimi~A.
\paper            On controlled frames in Hilbert $C^*$-modules
\jour             Int. J. Wavelets Multiresolut. Inf. Process.
\yr               2017
\vol              15
\issue            4
\num              1750038
\size             15
\endref

\ref\no 13
\by               Kutyniok~G.,  Okoudjou~K.A., Philipp~F., and Tuley~E.K.
\paper            Scalable frames
\jour             Linear Algebra Appl.
\yr               2013
\vol              438
\issue            5
\pages            2225--2238
\endref

\ref\no 14
\by               Casazza~P.G.,  De Carli~L., and Tran~T.T.
\paper            Piecewise scalable frames
\jour             Linear Algebra Appl.
\yr               2024
\vol              694
%\issue            -
\pages            262--282
\endref

\ref\no 15
\by               G\u avru\c ta~L.
\paper            Frames for operators
\jour             Appl. Comput. Harmon. Anal.
\yr               2012
\vol              32
\issue            1
\pages            139--144
\endref

\ref\no 16
\by               Jahan~S., Kumar~V., and Kaushik~S.K.
\paper            On the existence of non-linear frames
\jour             Arch. Math. (Brno)
\yr               2017
\vol              53
\issue            2
\pages            101--109
\endref

\ref\no 17
\by               Jahan~S.
\paper            Approximative $K$-atomic decompositions and frames in Banach spaces
\jour             Arab J. Math. Sci.
\yr               2020
\vol              26
\iftex
\issue            1--2
\else
\issue            1
\fi
\pages            153--166
\endref

\ref\no 18
\by                Alizadeh~E.,  Rahimi~A.,  Osgooei~E., and Rahmani~M.
\paper             Continuous $K$-$g$-fusion frames in Hilbert spaces
%\?в заголовке в статье $K$-$G$-fusion, но, думаю, просто заголовок весь большими буквами, а в статье всюду $K$-$g$-fusion
\jour              TWMS J. Appl. Eng. Math.
\yr                2021
\vol               11
\issue             1
\pages             44--55
\endref

\ref\no 19
\by                Xia~X.G. and  Suter~B.W.
\paper             Vector-valued wavelets and vector filter banks
\jour              IEEE Trans. Signal Process.
\yr                1996
\vol               44
\issue             3
\pages             508--518
\endref

\ref\no 20
\by                San Antol\'{\i}n~A. and Zalik~R.A.
\paper             Matrix-valued wavelets and multiresolution analysis
\jour              J. Appl. Funct. Anal.
\yr                2012
\vol               7
\iftex
\issue             1--2
\else
\issue             1
\fi
\pages             13--25
\endref

\ref\no 21
\by               Vashisht~L.K.
\paper            $\Cal K$-Matrix-valued wave packet frames in $L^2 (\Bbb{R}^d, \Bbb{C}^{s\times r})$
\jour             Math. Phys. Anal. Geom.
\yr               2018
\vol              21
\issue            3
\num              21
\size             19
\endref

\ref\no 22
\by               Jindal~D., Jyoti, and Vashisht~L.K.
\paper            Matrix-valued nonstationary frames associated with the Weyl--Heisenberg group and the extended affine group
\jour             Int. J. Wavelets Multiresolut. Inf. Process.
\yr               2023
\vol              21
\issue            6
\num              2350022
\size             25
\endref

\ref\no 23
\by               Wigner~E.
\paper            On the quantum correction for thermodynamic equilibrium
\jour             Phys. Rev.
\yr               1932
\vol              40
\issue            5
\pages            749
\endref

\ref\no 24
\by               Ville~J.
\paper            Th\'eorie et applications de la notion de signal analytique
\jour             C\'ables et Transmissions
\yr               1948
\vol              2
\issue            1
\pages            61--74
\endref

\ref\no 25
\by               Cordero~E. and  Rodino~L.
\paper            Wigner analysis of operators. Part I: Pseudodifferential operators and wave fronts
\jour             Appl. Comput. Harmon. Anal.
\yr               2022
\vol              58
%\issue           -
\pages            85--123
\endref

\ref\no 26
\by               Cordero~E. and Giacchi~G.
\paper            Metaplectic Gabor frames and symplectic analysis of time-frequency spaces
\jour             Appl. Comput. Harmon. Anal.
\yr               2024
\vol              68
%\issue           -
\num              101594
\size             31
\endref

\ref\no 27
\by               Cordero~E. and Giacchi~G.
\paper            Metaplectic Gabor frames of Wigner-decomposable distributions
\inbook           International Conference on Numerical Computations: Theory and Algorithms
\publaddr         Cham
\publ             Springer Nature Switzerland
\yr               2023
\pages            63--77
\endref

\ref\no 28
\by               Balazs~P., Antoine~J.-P., and Grybo\'s~A.
\paper            Weighted and controlled frames: Mutual relationship and first numerical properties
\jour             Int. J. Wavelets Multiresolut. Inf. Process.
\yr               2010
\vol              8
\issue            1
\pages            109--132
\endref

\ref\no 29
\by               Cordero~E. and  Rodino~L.
      \book       Time-Frequency Analysis of Operators
      \publ       De Gruyter
      \publaddr   Berlin
            \yr   2020
\finalinfo        De Gruyter Stud. Math., 75
\endref

\ref\no 30
\by               Vretblad~A.
      \book       Fourier Analysis and Its Applications
      \publ       Springer
      \publaddr   New York
            \yr   2003
\finalinfo        Grad. Texts in Math., 223
\endref
\endRefs

\enddocument