# Frames With Several Generators Associated with Weyl-Heisenberg Group and
Extended Affine Group

Divya Jindal  Divya Jindal, Department of Mathematics, University of Delhi,
Delhi-110007, India<EMAIL_ADDRESS>and  Lalit Kumar Vashisht
Lalit Kumar Vashisht, Department of Mathematics, University of Delhi,
Delhi-110007, India<EMAIL_ADDRESS>
###### Abstract.

We study the construction of Gabor frames and wavelet frames for Weyl-
Heisenberg group and extended affine group by using contraction between the
affine group and the Weyl-Heisenberg group due to Subag, Baruch, Birman and
Mann. Firstly, we give construction of Gabor frames with several generators
from a unitary irreducible representation associated to the Weyl-Heisenberg
group. Wavelet frames with several generators associated with an extended
affine group have been obtained. A relation between frames for Weyl-Heisenberg
group and extended affine group is also discussed. Finally, we show that
frames of Gabor and wavelet structure are stable under small perturbations.

††2020 _Mathematics Subject Classification_ : Primary 42C15; Secondary 42C30;
42C40.†† _Key words and phrases_ : Frames, Gabor frames; wavelet frames; Weyl-
Heisenberg group; Extended affine group; perturbation.
The research of Divya Jindal is supported by the Council of Scientific $\&$
Industrial Research (CSIR), India. Grant No.: 09/045(1680)/2019-EMR-I. Lalit
Kumar Vashisht is supported by the Faculty Research Programme Grant-IoE,
University of Delhi (Grant No.: Ref. No./IoE/2021/12/FRP).

##  1\. Introduction

Wavelet and Gabor analyses are perfect illustrations of the deep analogy
between quantum physics and signal processing, see [1, 10]. Gabor analysis
yields a time frequency representation of signals while wavelet provide a time
scale representation. More precisely, both Gabor analysis and wavelet analysis
are integral part of time-frequency analysis which gives uniform time and
frequency resolution. In recent years, reconstruction of signals and its
decomposition by using wavelet and Gabor systems have been extensively
studied, see [2, 4, 6, 9, 12, 13, 14, 18, 25, 26] and many references therein.
The wavelet and Gabor analyses are both induced by coherent states that are
associated with a unitary irreducible representation of a Lie group, see [1,
12, 14, 20] for details. Wavelet analysis corresponds to unitary irreducible
representations of the affine group while the Gabor analysis corresponds to
that of the Weyl-Heisenberg group. Let $\mathcal{H}$ be a nonzero separable
Hilbert space and $G$ be a locally compact group. A unitary representation of
$G$ is a homomorphism $\pi$ from $G$ into the group $\mathcal{U}(\mathcal{H})$
of unitary operators on $\mathcal{H}$ that is continuous with respect to the
strong operator topology, that is, a map $\pi:G\to\mathcal{U}(\mathcal{H})$
that satisfies $\pi(xy)=\pi(x)\pi(y)$ and
$\pi(x^{-1})=\pi(x)^{-1}=\pi(x)^{*}$, and for which $x\to\pi(x)u$ is
continuous from $G$ to $\mathcal{H}$ for any $u\in\mathcal{H}$. Also, $\pi$ is
said to be irreducible if $\pi$ admits only trivial (that is, $=\\{0\\}$ or
$\mathcal{H}$) invariant subspaces. A Lie group can be obtained as a certain
limit of another group with the help of group contraction, see [21] and
references therein. Let $G_{0}$ and $\widetilde{G_{0}}$ be two Lie groups. We
say that $\widetilde{G_{0}}$ is a _contraction_ of $G_{0}$ if for every
$\epsilon\in(0,1]$, there exists a diffeomorphism
$F_{\epsilon}:\widetilde{G_{0}}\to G_{0}$ such that
$F_{\epsilon}(e_{0})=e_{1}$ where $e_{0}$, $e_{1}$ are the identity elements
of $\widetilde{G_{0}}$ and $G_{0}$, respectively, and $x.y=\lim_{\epsilon\
to0^{+}}F_{\epsilon}^{-1}(F_{\epsilon}(x).F_{\epsilon}(y))$ for every
$x,y\in\widetilde{G_{0}}$. Very recently, Subag, Baruch, Birman, and Mann, in
[21], studied relation between Gabor and wavelet analyses by using
“contraction” between affine group and Weyl-Heisenberg group. They obtained
unitary irreducible representations of the Heisenberg group as contraction of
representations of the extended affine group. Using this technique, Subag et
al., in [21], relates two analyses, where they contracted coherent states,
tight frames and resolutions of the identity. In the direction of frame
conditions, Subag et al. constructed a family of time localized wavelet frames
that contract to a Gabor frame. In [21], Subag et al., also showed that how a
family of frequency localized wavelet frames can be contracted to non-standard
Gabor frames. They also gave deformation of Gabor frames to wavelet frames. To
connect wavelet and Gabor systems by special type of groups, the authors of
[5, 22, 23] gave some relations between the Gabor analysis and wavelet
analysis by exhibiting both the affine group and the Weyl-Heisenberg group as
subgroups of the four dimensional affine Weyl-Heisenberg group.

Motivated by above works which relates fundamental properties of frames from
wavelet systems to Gabor systems via contraction, we study construction of
frames with several generators for Weyl-Heisenberg group and extended affine
group. Subag et al., in [21], gave tight frames with Gabor and wavelet
structure associated with Weyl-Heisenberg group and extended affine group.
They obtain unitary irreducible representation of Weyl-Heisenberg group from
unitary irreducible representation of extended affine group using group
contraction. Using the representations given by Subag et al., in [21], we give
existence of Gabor frames and wavelet frames with several generators
associated with Weyl-Heisenberg group and extended affine group, respectively.

###  1.1. Outline and contributions

The paper is structured as follows: To make the paper self-contained, Section
2 gives a brief review on Gabor systems, wavelet systems, Weyl-Heisenberg
group and extended affine group. The main results are given in Section 3. Our
first result gives sufficient conditions for existence of Gabor frames with
several generators for the signal space $L^{2}(\mathbb{R},dx)$ from a unitary
irreducible representation associated to the Weyl-Heisenberg group, see
Theorem 3.2. The conditions given in Theorem 3.2 are only sufficient are
justified in Example 3.5. Theorem 3.7 gives wavelet frames with several
generators for associated with an extended affine group. A relation between
frames for Weyl-Heisenberg group and extended affine group can be found in
Theorem 3.8. Theorem 4.1 of Section 4 shows that frames of both Gabor and
wavelet structure are stable under small perturbations.

##  2\. Preliminaries

Symbols $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}^{+}$, $\mathbb{R}^{*}$ and
$\mathbb{R}$ stand for a set of the natural numbers, integers, positive real
numbers, non-zero real numbers and real numbers, respectively. To make
notations compatible with quantum physics, in this paper, a vector $\phi$ of a
Hilbert space $\mathbb{H}$ is also denoted by $|\phi\rangle$ (ket vector or
state in $\mathbb{H}$). The inner product on $\mathbb{H}$ is denoted by
$\langle\cdot|\cdot\rangle$. The inner product $\langle\cdot|\cdot\rangle$ is
conjugate-linear in the first variable and linear in the first variable, that
is,

$\displaystyle\langle\phi|\alpha\psi+\beta\varphi\rangle=\alpha\langle\phi|\psi\rangle+\beta\langle\phi|\varphi\rangle\
\ \text{and}\ \
\langle\alpha\phi|\psi\rangle=\overline{\alpha}\langle\phi|\psi\rangle,$

for all vectors $|\phi\rangle$, $|\psi\rangle$ and $|\varphi\rangle$ in
$\mathbb{H}$ and for all scalars $\alpha$, $\beta$. Here, $\overline{\alpha}$
denote the complex-conjugate of the scalar $\alpha$. The norm on $\mathbb{H}$,
denoted by $\|\cdot\|$, is defined as $\|\phi\|=\langle\phi|\phi\rangle$ for
$|\phi\rangle\in\mathbb{H}$. As is standard, the space of equivalence classes
of square integrable (Lebesgue integrable) functions is denoted by
$L^{2}(\mathbb{R},dx)$. Here $dx$ is the Lebesgue measure. Engineers and
physicists pronounced $L^{2}(\mathbb{R},dx)$ as the “ _signal space_ ”. We
consider the following standard inner product on $L^{2}(\mathbb{R},dx)$:

$\displaystyle\langle\phi|\psi\rangle=\int_{\mathbb{R}}\overline{\phi(x)}\psi(x)dx,\
\ \phi,\,\psi\in L^{2}(\mathbb{R},dx).$ (2.1)

The space $L^{2}(\mathbb{R},dx)$ becomes a Hilbert space with respect to the
inner product defined in (2.1).

To define Gabor and wavelet system, the following classes of operators, which
acts unitarily on $L^{2}(\mathbb{R},dx)$, are used: Let $a$, $b$ be positive
real numbers, $c$ be a non-zero real number and let $\phi$ be a non-zero
vector in $L^{2}(\mathbb{R})$.

$\displaystyle\ T_{a}:\phi(x)\mapsto\phi(x-a)\ \quad(\text{Translation by}\
a);$ $\displaystyle E_{b}:\phi(x)\mapsto e^{2\pi ib\cdot
x}\phi(x)\quad(\text{Modulation by}\ b);$ $\displaystyle
D_{c}:\phi(x)\mapsto|c|^{\frac{1}{2}}\phi(cx)\ \ \,\quad(\text{Dilation by}\
c).$

A collection of vectors of the form
$\mathcal{G}(a,b,\phi):=\\{E_{mb}T_{na}\phi\\}_{m,n\in\mathbb{Z}}=\\{e^{2\pi
imbx}\phi(x-a)\\}_{m,n\in\mathbb{Z}}$ is called a _Gabor system_ ; and the
collection
$\mathcal{W}(a,b,\phi):\\{T_{kba^{j}}D_{a^{j}}\phi\\}_{j,k\in\mathbb{Z}}=\\{a^{\frac{j}{2}}\phi(a^{j}x-kb)\\}_{j,k\in\mathbb{Z}}$
is called a _wavelet system_.

###  2.1. Hilbert Frames

Dennis Gabor, in [11], introduced a fundamental approach to signal
decomposition in terms of elementary signal, for example, the Gaussian. Duffin
and Schaeffer [8] while studying some intense problems related to non-harmonic
Fourier series, abstracted the fundamental notion of Gabor to introduce the
notion of a frame for the signal space $L^{2}((-\beta,\beta),dx)$. To be
precise, they defined frames of exponentials for the space
$L^{2}((-\beta,\beta),dx)$. Gabor frames or Weyl-Heisenberg frames originate
in quantum mechanics by Neumann [19], and in communication theory by the idea
of Gabor, see [11]. A countable collection of vectors
$\\{|w_{k}\rangle\\}_{k\in\mathfrak{I}}$ in a separable Hilbert space
$\mathbb{H}$ is a _Hilbert frame_ (or simply, _frame_) for $\mathbb{H}$ if
there exist positive real numbers $\gamma_{o}$, $\delta_{o}$ such that

$\displaystyle\gamma_{o}\|w\|^{2}\leq\sum_{k\in\mathfrak{I}}|\langle
w,w_{k}\rangle|^{2}\leq\delta_{o}\|w\|^{2}$ (2.2)

holds for all $|w\rangle$ in $\mathbb{H}$. The scalars $\gamma_{o}$,
$\delta_{o}$ are called lower and upper frame bounds of the frame
$\\{w_{k}\\}_{k\in\mathfrak{I}}$. The frame
$\\{|w_{k}\rangle\\}_{k\in\mathfrak{I}}$ _frame_ is tight if it is possible to
choose $\alpha_{o}=\delta_{o}$, and _normalized tight_ (or _Parseval_ , if
$\alpha_{o}=\delta_{o}=1$. If only upper inequality holds in (2.2), then we
say that $\\{|w_{k}\rangle\\}_{k\in\mathfrak{I}}$ is a _Bessel sequence_ with
_Bessel bound_ $\delta_{o}$. If $\\{|w_{k}\rangle\\}_{k\in\mathfrak{I}}$ is a
frame for $\mathbb{H}$, then the map
$\mathrm{S}:\mathcal{H}\rightarrow\mathcal{H}$, known as the _frame operator_
, given by $\mathrm{S}:|w\rangle\mapsto\sum_{k\in\mathfrak{I}}\langle
w,w_{k}\rangle w_{k}$ is linear, bounded and invertible on $\mathbb{H}$. Thus,
a frame for $\mathbb{H}$ provides a decomposition, not necessarily unique, for
every vector $|w\rangle$ of $\mathbb{H}$:

$\displaystyle|w\rangle=\mathrm{S}\mathrm{S}^{-1}|w\rangle=\sum_{k\in\mathfrak{I}}\big{(}\langle\mathrm{S}^{-1}w,|w_{k}\rangle\big{)}\,|w_{k}\rangle.$

This decomposition turns out to be a powerful tool for deriving many
fundamental results in abstract frame theory. Further, it plays central role
both in both pure mathematics and engineering science, for technical details
about this, we refer to excellent texts by Casazza and Kutyniok [2] and Heil
[13]. Nowadays the theory of wavelets, distributed signal processing, quantum
mechanics, iterated function systems and many more topics are emerging in
important applications in frame theory, see [7, 15, 16, 24] and many
references therein. A frame of the form $\mathcal{G}(a,b,\phi)$ is called the
Gabor frame and wavelet frame is a frame of the form $\mathcal{W}(a,b,\phi)$.

###  2.2. The Weyl-Heisenberg group

The Weyl-Heisenberg group is the semidirect product
$\mathbb{R}^{2}\rtimes_{\psi}\mathbb{R}$, where $\psi:\mathbb{R}\rightarrow
Aut(\mathbb{R}^{2})$ is given by $\psi_{\lambda}((x_{1},x_{2}))=(x_{1}+\lambda
x_{2},x_{2})$ for every $\lambda,x_{1},x_{2}\in\mathbb{R}$; and denote
$\mathcal{W}=\mathbb{R}^{2}\rtimes_{\psi}\mathbb{R}$ with multiplication law

$\displaystyle((x_{1},x_{2}),\lambda)((y_{1},y_{2}),\nu)=(\psi_{\lambda}(y_{1},y_{2})+(x_{1},x_{2}),\lambda+\nu).$

Recall that for every $P\in\mathbb{R}^{*},Q\in\mathbb{R}$, the unitary
irreducible representation of this group acting on $\mathcal{H}$,
$\kappa^{P,Q}:\mathcal{W}\rightarrow\mathcal{U}(L^{2}(\mathbb{R},dx))$ is
given by

$\displaystyle(\kappa^{P,Q}(((u_{1},u_{2}),z))f)(x)=e^{i[P(u_{1}+xu_{2})+Qu_{2}]}f(z+x).$

For $l\in\\{1,2,\dots,N\\}$, where $N$ is some fixed natural number, let
$q_{0}^{(l)}$ and $p_{0}^{(l)}$ be real numbers such that
$|q_{0}^{(l)}p_{0}^{(l)}|<2\pi$. Consider the discrete subset of $\mathcal{W}$
given by

$\displaystyle\mathcal{W}_{q_{0}^{(l)},p_{0}^{(l)}}=\bigg{\\{}\bigg{(}\bigg{(}\frac{mnq_{0}^{(l)}p_{0}^{(l)}}{2},mp_{0}^{(l)}\bigg{)},nq_{0}^{(l)}\bigg{)}|n,m\in\mathbb{Z}\bigg{\\}}.$

For $0\neq\phi_{l}\in L^{2}(\mathbb{R},dx)$ where $l\in\\{1,2,\dots,N\\}$, the
sequence

$\displaystyle\big{|}\phi_{(m,n,l)}^{P,Q}\big{>}=\kappa^{P,Q}\bigg{(}\bigg{(}\frac{mnq_{0}^{(l)}p_{0}^{(l)}}{2},mp_{0}^{(l)}\bigg{)},nq_{0}^{(l)}\bigg{)}\phi_{l},$

where $n,m\in\mathbb{Z}$ is dense in $L^{2}(\mathbb{R},dx)$, see [21] for
technical details.

###  2.3. Extended Affine Group

The affine group $A$ realized as the upper half plane

$A=\\{(a,b):a\in\mathbb{R}^{+},b\in\mathbb{R}\\}$,

and the multiplication law in $A$ is given by

$(a_{1},b_{1})(a_{2},b_{2})=(a_{1}a_{2},a_{1}b_{2}+b_{1})$,

for all points $(a_{1},b_{1}),(a_{2},b_{2})\in A$.

The support of a function $f$, denoted by supp$f$, is defined as supp$f:=$
closure of the set $\\{x:f(x)\neq 0\\}$. Let $V$ be a subspace of
$L^{2}(\mathbb{R},dx)$ given by

$\displaystyle V=\Big{\\{}f\in
L^{2}(\mathbb{R},dx):\mbox{supp}(\hat{f})\subseteq[0,\infty)\Big{\\}}$ (2.3)

where $\hat{f}$ is the Fourier transform of $f$ given by

$\displaystyle\hat{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{-iwx}dx.$

Recall that for any $\alpha\in\mathbb{R},$ the unitary irreducible
representation of the affine group on the subspace $V$ of
$L^{2}(\mathbb{R},dx)$, as given in (2.3), is the map
$\eta^{\alpha}:A\to\mathcal{U}(V)$ given by

$(\eta^{\alpha}(a,b)f)(x)=\frac{1}{\sqrt{a}}f(\frac{x+\alpha
b}{a}),\,x\in\mathbb{R}$;

and correspondingly define the sequence

$\phi^{\alpha}_{a,b}(x)=(\eta^{\alpha}(a,b)\psi)(x)=\frac{1}{\sqrt{a}}\psi(\frac{x+\alpha
b}{a})$.

The extended affine group, denoted by $EA$, is given by the direct product of
the affine group $A$ with additive group of real numbers. That is,
$EA=A\oplus\mathbb{R}$. The multiplication law on $EA$ is given by

$(a,b,y)(c,d,z)=(ac,ad+b,y+z)$.

For $\beta\in\mathbb{R},$ let $\chi_{\beta}(x)=e^{i\beta x}$ be unitary
character of $\mathbb{R}.$ For every $\beta\in\mathbb{R}$,
$\alpha\in\mathbb{R^{*}}$, the unitary irreducible representation of $EA$
acting on $V$ is the map $\eta^{\alpha}\otimes\chi_{\beta}:EA\to\mu(V)$ given
by

$(\eta^{\alpha}\otimes\chi_{\beta}(a,b,y))f)(x)=e^{i\beta
y}\frac{1}{\sqrt{a}}f(\frac{x+\alpha b}{a})$.

For finding the relation between frames for $EA$ and frames for $W$, we need a
different realization which is better suited. This realization is given by
three linear invertible maps $S_{1}$, $S_{2}$, $S_{3}(\epsilon)$ defined by

  1. $(i)$

$S_{1}:V\to L^{2}(\mathbb{R}^{+},dx)$,
$S_{1}(f)(x)=\hat{f}(x)|_{\mathbb{R}^{+}}$.

  2. $(ii)$

$S_{2}:L^{2}(\mathbb{R}^{+},dx)\to L^{2}(\mathbb{R}^{+},\frac{1}{x}dx)$,
$S_{2}(f)(x)=\sqrt{x}f(x)$.

  3. $(iii)$

$S_{3}(\epsilon):L^{2}(\mathbb{R}^{+},\frac{1}{x}dx)\to
L^{2}(\mathbb{R}^{+},dx)$, $S_{3}(\epsilon)(f)(x)=f\circ\psi_{\epsilon}(x)$,
where $\psi_{\epsilon}(x)=e^{-\epsilon x}$.

Set $U_{\epsilon}=S_{3}(\epsilon)\circ S_{2}\circ S_{1}$. For each
$\epsilon\in(0,1]$, $\alpha\in\mathbb{R}^{*}$ and $\beta\in\mathbb{R}$,
interwine $\eta^{\alpha}\otimes\chi_{\beta}$ with $U_{\epsilon}$ to get the
equivalent representation

$\zeta^{\beta,\alpha}_{\epsilon}:EA\to\mathcal{U}(V)$

$\displaystyle\begin{split}(\zeta^{\beta,\alpha}_{\epsilon}(a,b,y)f)|_{x}&=U_{\epsilon}\circ(\eta^{\alpha}\otimes\chi_{\beta})(a,b,y)\circ
U_{\epsilon})(f)(x)\\\ &=e^{i\beta y}e^{i\alpha be^{-\epsilon x}}f(\frac{-\log
a}{\epsilon}+x).\end{split}$

##  3\. Main Results

We begin this section with the following definition.

######  Definition 3.1.

Let $N$ be a fixed natural number. A collection of vectors
$\\{\big{|}\phi_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ in $\mathcal{H}$ is said to be a _frame_ for
$\mathcal{H}$ if there exist positive real constants $\alpha_{o}$ and
$\beta_{o}$ such that

$\displaystyle\alpha_{o}\|f\|^{2}\leq\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi_{(m,n,l)}\rangle\|^{2}\leq\beta_{o}\|f\|^{2}\ \text{for all}\
f\in\mathcal{H}.$ (3.1)

The scalars $\alpha_{o}$, $\beta_{o}$, obviously not unique, are called _lower
frame bound_ and _upper frame bound_ , respectively, for
$\\{\big{|}\phi_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$.

###  3.1. Frames for Weyl-Heisenberg Group

The following theorem gives sufficient conditions for a sequence of functions
obtained from a unitary irreducible representation associated to the Weyl-
Heisenberg group to be a Gabor frame with several generators.

######  Theorem 3.2.

For each $l\in\\{1,2,\dots,N\\}$, let $\phi_{l}$ be a non-zero function in
$L^{2}(\mathbb{R},dx)$ such that support of each $\phi_{l}$ is contained in an
interval of length $\mu_{l}$. Assume that

  1. $(i)$

$p_{0}^{(l)}=\frac{2\pi}{P\mu_{l}}$,

  2. $(ii)$

$|q_{0}^{(l)}p_{0}^{(l)}|<2\pi$, that is, $|q_{0}^{(l)}|<|P|\mu_{l}$ for some
fixed $q_{0}^{(l)}$,

  3. $(iii)$

For each $l\in\\{1,2,\dots,N\\}$,

$\displaystyle\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\leq\sum_{n\in\mathbb{Z}}\big{|}\phi_{l}\big{(}x-nq_{0}^{(l)}\big{)}\big{|}^{2}\leq\frac{\beta_{o}}{N\max\limits_{1\leq
i\leq N}\mu_{i}},\ \ \text{for almost all}\ x\in\mathbb{R}.$ (3.2)

Then, the collection
$\\{\big{|}\phi_{(m,n,l)}^{P,Q}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ is a multivariate frame for $L^{2}(\mathbb{R},dx)$
with frame bounds $\alpha_{o}$, $\beta_{o}$.

###### Proof.

Let $f\in L^{2}(\mathbb{R},dx)$ be arbitrary. By invoking the Parseval’s
theorem, we compute

$\displaystyle{}\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi_{(m,n,l)}^{P,Q}\rangle|^{2}$
$\displaystyle=\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}\bigg{|}\int_{\mathbb{R}}\overline{f(x)}e^{iPx\frac{2\pi}{P\mu_{l}}m}\phi_{l}(x+nq_{0}^{(l)})dx\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}\bigg{|}\int_{\mathbb{R}}\overline{f(y-nq_{0}^{(l)})}e^{i(y-nq_{0}^{(l)})\frac{2\pi}{\mu_{l}}m}\phi_{l}(y)dy\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}\bigg{|}\int_{\mathbb{R}}\overline{f(y-nq_{0}^{(l)})}e^{iy\frac{2\pi}{\mu_{l}}m}\phi_{l}(y)dy\bigg{|}^{2}$
$\displaystyle=\mu_{1}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(y-nq_{0}^{(1)})\phi_{1}(x)|^{2}dx+\mu_{2}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(y-nq_{0}^{(2)})\phi_{2}(x)|^{2}dx$
$\displaystyle+\dots+\mu_{N}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(y-nq_{0}^{(N)})\phi_{N}(x)|^{2}dx$
$\displaystyle=\mu_{1}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(x)|^{2}|\phi_{1}(x+nq_{0}^{(1)})|^{2}dx+\mu_{2}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(x)|^{2}|\phi_{2}(x+nq_{0}^{(2)})|^{2}dx$
$\displaystyle+\dots+\mu_{N}\sum_{n\in\mathbb{Z}}\int_{\mathbb{R}}|f(x)|^{2}|\phi_{N}(x+nq_{0}^{(N)})|^{2}dx$
$\displaystyle=\mu_{1}\int_{\mathbb{R}}|f(x)|^{2}\sum_{n\in\mathbb{Z}}|\phi_{1}(x+nq_{0}^{(1)})|^{2}dx+\mu_{2}\int_{\mathbb{R}}|f(x)|^{2}\sum_{n\in\mathbb{Z}}|\phi_{2}(x+nq_{0}^{(2)})|^{2}dx$
$\displaystyle+\dots+\mu_{N}\int_{\mathbb{R}}|f(x)|^{2}\sum_{n\in\mathbb{Z}}|\phi_{N}(x+nq_{0}^{(N)})|^{2}dx$
$\displaystyle\geq\mu_{1}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx+\mu_{2}\frac{\alpha_{o}}{N\min\limits_{1\leq
i\leq N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx$
$\displaystyle+\dots+\mu_{N}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx$
$\displaystyle\geq\mu_{1}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\mu_{2}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\dots+\mu_{N}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}$ $\displaystyle\geq\min\limits_{1\leq i\leq
N}\mu_{i}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\min\limits_{1\leq i\leq
N}\mu_{i}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq N}\mu_{i}}\|f\|^{2}$
$\displaystyle+\dots+\min\limits_{1\leq i\leq
N}\mu_{i}\frac{\alpha_{o}}{N\min\limits_{1\leq i\leq N}\mu_{i}}\|f\|^{2}$
$\displaystyle=\alpha_{o}\|f\|^{2}.$ (3.3)

Similarly, for any $f\in L^{2}(\mathbb{R},dx)$, we have

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi_{(m,n,l)}^{P,Q}\rangle|^{2}$
$\displaystyle\leq\mu_{1}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx+\mu_{2}\frac{\beta_{o}}{N\max\limits_{1\leq
i\leq
N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx+\dots+\mu_{N}\frac{\beta_{o}}{N\max\limits_{1\leq
i\leq N}\mu_{i}}\int_{\mathbb{R}}|f(x)|^{2}dx$
$\displaystyle\leq\mu_{1}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\mu_{2}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\dots+\mu_{N}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}$ $\displaystyle\leq\max\limits_{1\leq i\leq
N}\mu_{i}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\max\limits_{1\leq i\leq
N}\mu_{i}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq
N}\mu_{i}}\|f\|^{2}+\dots+\max\limits_{1\leq i\leq
N}\mu_{i}\frac{\beta_{o}}{N\max\limits_{1\leq i\leq N}\mu_{i}}\|f\|^{2}$
$\displaystyle=\beta_{o}\|f\|^{2}.$ (3.4)

Inequalities $(\ref{h1})$ and $(\ref{h2})$ concludes the result. ∎

######  Remark 3.3.

For $N>1$, the converse of Theorem 3.2 is not true. This is justified in
Example 3.5, where following result is used.

######  Theorem 3.4.

[3, Theorem 11.4.2] Let $a$, $b>0$ and $g$ be a non-zero function in
$L^{2}(\mathbb{R},dx)$. Suppose that

$\displaystyle\beta_{o}:=\frac{1}{b}\sup_{x\in[0,a]}\sum_{k\in\mathbb{Z}}\bigg{|}\sum_{n\in\mathbb{Z}}g(x-na)\overline{g(x-na-k/b)}\bigg{|}<\infty,$
and
$\displaystyle\gamma_{o}:=\frac{1}{b}\inf_{x\in[0,a]}\biggr{[}\sum_{n\in\mathbb{Z}}|g(x-na)|^{2}-\sum_{k\neq
0}\bigg{|}\sum_{n\in\mathbb{Z}}g(x-na)\overline{g(x-na-k/b)}\bigg{|}\biggr{]}>0.$

Then, $\\{e^{2\pi imbx}g(x-na)\\}_{m,n\in\mathbb{Z}}$ is a frame for
$L^{2}(\mathbb{R},dx)$ with frame bounds $\gamma_{o}$, $\beta_{o}$.

Now, we show that, for $N>1$, the converse of Theorem 3.2 is not true.

######  Example 3.5.

Let $P=1$, $Q=0$ and define a function $\phi_{1}\in L^{2}(\mathbb{R},dx)$,
with $\mu_{1}=2$, as follows:

$\displaystyle\phi_{1}(x)=\begin{cases}1+x,\quad&\text{if}\ x\in(0,1];\\\
x,\quad&\text{if}\ x\in(1,2];\\\ 0,\quad&\text{elsewhere}.\end{cases}$

Choose $q_{0}^{(1)}=1$, $a=1$ and $b=\frac{1}{\mu_{1}}=\frac{1}{2}$. Consider
for $n$, $k\in\mathbb{Z}$, the function
$x\rightarrow\phi_{1}(x-n)\phi_{1}(x-n-2k)$ for $x\in(0,1]$. Due to the
compact support of $\phi_{1}$, it can only be non-zero if $n\in\\{-1,0\\}$ and
$k=0$. So,

$\displaystyle
G_{0}(x)=\sum_{n\in\mathbb{Z}}|g(x-n)|^{2}=|g(x)|^{2}+|g(x+1)|^{2}=2(x^{2}+2x+1)\
\text{for}\ x\in(0,1],$ and $\displaystyle G_{1}(x)=\sum_{k\neq
0}\bigg{|}\sum_{n\in\mathbb{Z}}g(x-n)\overline{g(x-na-2k)}\bigg{|}=0\
\text{for}\ x\in(0,1].$

Therefore, by Theorem 3.4,
$\\{e^{2ix\frac{m}{\mu_{1}}\pi}\phi_{1}(x+n)\\}_{m,n\in\mathbb{Z}}$ is a frame
for $L^{2}(\mathbb{R},dx)$ with frame bounds $4$ and $16$. Hence,
$\big{\\{}\big{|}\phi_{(m,n,1)}^{1,0}\big{>}\big{\\}}_{m,n\in\mathbb{Z}}$ is a
frame for $L^{2}(\mathbb{R},dx)$ with frame bounds $4$ and $16$.

Similarly, for the following function $\phi_{2}$ in $L^{2}(\mathbb{R},dx)$,

$\displaystyle\phi_{2}(x)=\begin{cases}1+x,\quad&\text{if}\ x\in(0,1],\\\
\frac{x}{2},\quad&\text{if}\ x\in(1,2],\\\
0,\quad&\text{elsewhere}.\end{cases}$

with $q_{0}^{(2)}=1,a=1$ and $b=\frac{1}{\mu_{2}}=\frac{1}{2}$, the sequence
$\big{\\{}\big{|}\phi_{(m,n,2)}^{1,0}\big{>}\big{\\}}_{m,n\in\mathbb{Z}}$ is a
frame for $L^{2}(\mathbb{R},dx)$ with frame bounds $\frac{1}{4}$ and $9$.
Thus, for all $f\in L^{2}(\mathbb{R},dx)$, we have

$\displaystyle\frac{17}{4}\|f\|^{2}\leq\sum_{l=1}^{2}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi_{(m,n,l)}^{1,0}\rangle|^{2}\leq 25\|f\|^{2}.$

Hence, the sequence
$\\{\big{|}\phi_{(m,n,l)}^{1,0}\big{>}\\}_{m,n\in\mathbb{Z},l\in\\{1,2\\}}$ is
a multivariate frame with bound $\frac{17}{4}$ and $25$.

One may observe that for $x=\sqrt{3.5}$, we have
$\sum_{n\in\mathbb{Z}}|\phi_{1}(x-n)|^{2}=7$. But this is not possible as
inequality (3.2) implies that

$\displaystyle\frac{17}{16}\leq\sum\limits_{n\in\mathbb{Z}}|\phi_{l}(x-n)|^{2}\leq\frac{25}{4}\
\ \text{for}\ l\in\\{1,2\\},\ \text{for almost all}\ x\in\mathbb{R}.$

######  Remark 3.6.

The converse of Theorem 3.2 is true if inequality (3.2) is replaced by

$\displaystyle\alpha_{o}\leq\sum_{l=1}^{N}\sum_{n\in\mathbb{Z}}\mu_{l}\big{|}\phi_{l}\big{(}x-nq_{0}^{(l)}\big{)}\big{|}^{2}\leq\beta_{o},\
\text{for almost all}\ x\in\mathbb{R}.$

###  3.2. Frames for Extended Affine Group

In this section, we give existence of wavelet frames with several generators
associated with extended affine group. Here, we fix a representation
$\kappa^{P,Q}$ of the Weyl-Heisenberg group on $L^{2}(\mathbb{R})$; and let
$u(\epsilon)=u_{0}+\frac{P}{\epsilon}$ and
$v(\epsilon)=v_{0}-\frac{P}{\epsilon}$ be such that $u_{0}+v_{0}=Q$. Define a
sequence on extended affine group, $EA$, by

$\displaystyle\big{|}\phi_{(m,n,l)}^{\epsilon,u(\epsilon),v(\epsilon)}\big{>}=\zeta_{\epsilon}^{u(\epsilon),v(\epsilon)}(a,b,y)\phi_{l},$

where $m,n\in\mathbb{Z}$ and $l\in\\{1,2,\dots,N\\}$ and the symbol
$\zeta_{\epsilon}^{\bullet,\bullet}(.,.,.)$ is defined in Section 2.3. The
following result gives multivariate wavelet frames for signal space
$L^{2}(\mathbb{R},dx)$.

######  Theorem 3.7.

Under the assumptions of Theorem 3.2, for any $\epsilon\in(0,1]$, let
$EA_{q_{0}(l),p_{0}(l)}(\epsilon)$ be a discrete subset of $EA$ given by

$\displaystyle\Big{\\{}Q_{n}^{(l)}(\epsilon),b_{mn}^{(l)}(\epsilon),y_{mn}^{(l)}(\epsilon)|\
m,n\in\mathbb{Z},l\in\\{1,2,\dots,N\\}\Big{\\}},$

where

$\displaystyle a_{n}^{(l)}(\epsilon)$ $\displaystyle=e^{-\epsilon
nq_{0}^{(l)}},$ $\displaystyle b_{mn}^{(l)}(\epsilon)$
$\displaystyle=\frac{\pi
a_{n}^{(l)}(\epsilon)m\beta_{o}}{\alpha_{o}v(\epsilon)(ln(\epsilon+c)-lnc)c\mu_{l}},$
$\displaystyle=\frac{\beta_{o}e^{-\epsilon nq_{0}^{(l)}}\pi
m}{\alpha_{o}v(\epsilon)(ln(\epsilon+c)-lnc)c\mu_{l}}$ $\displaystyle
y_{mn}^{(l)}(\epsilon)$
$\displaystyle=b_{mn}(\epsilon)\frac{l_{n}a_{n}^{(l)}(\epsilon)}{a_{n}^{(l)}(\epsilon)-1}.$

Then, the collection
$\big{\\{}\big{|}\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}=\Big{\\{}\zeta_{\epsilon}^{u(\epsilon),v(\epsilon)}(a_{n}^{(l)}(\epsilon),\beta_{mn}^{(l)}(\epsilon),\gamma_{mn}^{(l)}(\epsilon))(T_{\epsilon}\phi)\Big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$, where $T_{\epsilon}(x)=e^{\frac{-1}{2}\epsilon x}$,
forms a frame for $L^{2}(\mathbb{R},dx)$ with frame bounds

$\displaystyle\frac{\alpha_{o}^{2}c}{\beta_{o}}\Big{(}\frac{ln(\epsilon+c)-lnc}{\epsilon}\Big{)}\
\quad\text{and}\
\quad\alpha_{o}c\Big{(}\frac{ln(\epsilon+c)-lnc}{\epsilon}\Big{)}.$

###### Proof.

For any $f\in L^{2}(\mathbb{R},dx)$, we compute

$\displaystyle\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\big{\langle}f|\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{\rangle}\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{\mathbb{R}}\overline{f(x)}e^{iv(\epsilon)b_{mn}^{(l)}(\epsilon)e^{-\epsilon
x}}e^{\frac{-\epsilon}{2}(x-\frac{lna_{n}^{(l)}(\epsilon)}{\epsilon})}\phi_{l}(x-\frac{lna_{n}^{(l)}(\epsilon)}{\epsilon})dx\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{\mathbb{R}}\overline{f(x)}e^{iv(\epsilon)b_{mn}^{(l)}(\epsilon)e^{-\epsilon
x}}e^{\frac{-\epsilon}{2}(x+nq_{0}^{(l)})}\phi_{l}(x+nq_{0}^{(l)})dx\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{\mathbb{R}^{+}}\overline{f(\frac{-\ln{s}}{\epsilon})}e^{iv(\epsilon)b_{mn}^{(l)}(\epsilon)s}e^{\frac{-\epsilon}{2}(\frac{-\ln{s}}{\epsilon}+nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{s}}{\epsilon}+nq_{0}^{(l)})\frac{ds}{\epsilon
s}\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{\mathbb{R}^{+}}\overline{f(\frac{-\ln{s}}{\epsilon})}e^{iv(\epsilon)b_{mn}^{(l)}(\epsilon)s}e^{\frac{-\epsilon}{2}(nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{s}}{\epsilon}+nq_{0}^{(l)})\frac{ds}{\epsilon\sqrt{s}}\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{E_{l}}\overline{f(\frac{-\ln{t(a_{n}^{(l)}(\epsilon))}^{-1}}{\epsilon})}e^{iv(\epsilon)b_{mn}^{(l)}(\epsilon)\frac{t}{a_{n}^{(l)}(\epsilon)}}e^{\frac{-\epsilon}{2}(nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{t}}{\epsilon})\frac{dt}{\epsilon\sqrt{a_{n}(\epsilon)t}}\bigg{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\int_{E_{l}}\overline{f(\frac{-\ln{t(a_{n}^{(l)}(\epsilon))}^{-1}}{\epsilon})}e^{\frac{iB_{o}\pi
m}{A_{o}(\ln{\epsilon+c}-\ln{c})\mu_{l}c}}e^{\frac{-\epsilon}{2}(nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{t}}{\epsilon})\frac{dt}{\epsilon\sqrt{a_{n}(\epsilon)t}}\bigg{|}^{2}$
$\displaystyle=(\star\star),$

where $E_{l}\subseteq\mathbb{R}$ with finite measure depending on $\mu_{l}$
$(l\in\\{1,2,\dots,N\\})$.

By invoking the Parseval’s theorem, we get that

$\displaystyle(\star\star)$
$\displaystyle=\sum_{l=1}^{N}\frac{\alpha_{o}c\mu_{l}(\ln{\epsilon+c}-\ln{c})}{B_{o}}\underset{n\in\mathbb{Z}}{\sum}\int_{E_{l}}\bigg{|}\overline{f(\frac{-\ln{t(a_{n}^{(l)}(\epsilon))}^{-1}}{\epsilon})}e^{\frac{-\epsilon}{2}(nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{t}}{\epsilon})\frac{1}{\epsilon\sqrt{a_{n}(\epsilon)t}}\bigg{|}^{2}dt$
$\displaystyle=\sum\sum_{l=1}^{N}\frac{\alpha_{o}c\mu_{l}(\ln{\epsilon+c}-\ln{c})}{\beta_{o}\epsilon}\underset{n\in\mathbb{Z}}{\sum}\int_{\mathbb{R}^{+}}\bigg{|}\overline{f(\frac{-\ln{t(a_{n}^{(l)}(\epsilon))}^{-1}}{\epsilon})}e^{\frac{-\epsilon}{2}(nq_{0}^{(l)})}\phi_{l}(\frac{-\ln{t}}{\epsilon})\bigg{|}^{2}\frac{1}{\epsilon
t}dt.$ (3.5)

Change of variables $s=\frac{t}{a_{n}^{(l)}(\epsilon)}$ and then
$x=\frac{-\ln{s}}{\epsilon}$, in Eq. (3.2), gives

$\displaystyle\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}\big{|}\big{\langle}f|\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{\rangle}\big{|}^{2}$
$\displaystyle=\sum_{l=1}^{N}\frac{\alpha_{o}(\ln{(\epsilon+c})-\ln{c})c\mu_{l}}{\beta_{o}\epsilon}\underset{n\in\mathbb{Z}}{\sum}\int_{\mathbb{R}^{+}}\bigg{|}\overline{f(\frac{-\ln{s}}{\epsilon})}\phi_{l}(\frac{-\ln{s}a_{n}^{(l)}(\epsilon)}{\epsilon})\bigg{|}^{2}\frac{1}{\epsilon
s}ds$
$\displaystyle=\sum_{l=1}^{N}\frac{\alpha_{o}(\ln{(\epsilon+c})-\ln{c})c\mu_{l}}{B_{o}\epsilon}\underset{n\in\mathbb{Z}}{\sum}\int_{\mathbb{R}^{+}}\big{|}\overline{f(x)}\phi_{l}(x+nq_{0}^{(l)})\big{|}^{2}dx.$
(3.6)

Using Eq. (3.2), we have

$\displaystyle\frac{\alpha_{o}^{2}(ln{(\epsilon+c)}-\ln{c})c||f||^{2}}{\beta_{o}\epsilon}\leq\sum_{l\in\\{1,2,\dots,N\\}}\underset{m,n\in\mathbb{Z}}{\sum}\bigg{|}\big{\langle}f|\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{\rangle}\bigg{|}^{2}\leq\frac{\alpha_{o}(ln{(\epsilon+c)}-\ln{c})c||f||^{2}}{\epsilon},$

for all $f\in L^{2}(\mathbb{R},dx)$. Hence,
$\\{\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\\}_{m,n\in\mathbb{Z},l\in\\{1,2,\dots,N\\}}$
is a frame for $L^{2}(\mathbb{R},dx)$ with the desired frame bounds. ∎

Next theorem gives us the relation between frames with several generators for
Weyl-Heisenberg group and extended affine group.

######  Theorem 3.8.

Under the assumptions of Theorem 3.2, for any $f\in L^{2}(\mathbb{R},dx)$, we
have

$\displaystyle\lim_{\epsilon\to
0^{+}}\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}|\langle
f,\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\rangle|^{2}\leq\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}|\langle
f,\phi^{P,Q}_{(m,n,l)}\rangle|^{2}\leq\beta_{o}||f||^{2}$

###### Proof.

For any $f\in L^{2}(\mathbb{R},dx)$, we have

$\displaystyle\lim_{\epsilon\to
0^{+}}\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}|\langle
f,\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\rangle|^{2}$
$\displaystyle\leq\lim_{\epsilon\to
0^{+}}\frac{\alpha_{o}(\ln{(\epsilon+c)}-\ln{c})c||f||^{2}}{\epsilon}$
$\displaystyle=\lim_{\epsilon\to
0^{+}}\frac{\alpha_{o}c}{\epsilon+c}||f||^{2}$
$\displaystyle=\alpha_{o}||f||^{2}$
$\displaystyle\leq\sum_{l=1}^{N}\underset{m,n\in\mathbb{Z}}{\sum}|\langle
f,\phi^{P,Q}_{(m,n,l)}\rangle|^{2}$ $\displaystyle\leq\beta_{o}||f||^{2}.$

This concludes the proof. ∎

The following example illustrates Theorem 3.8.

######  Example 3.9.

Consider the Weyl-Heisenberg frame
$\big{\\{}\big{|}\phi_{(m,n,1)}^{1,0}\big{>}\big{\\}}_{m,n\in\mathbb{Z}}$ for
$L^{2}(\mathbb{R},dx)$ with frame bounds $4$ and $16$ given in Example 3.5.
Let $u_{0}=-1,v_{0}=1$ and $c=1$ be a fixed constant.

Define

$\displaystyle a_{n}^{(1)}(\epsilon)$ $\displaystyle=e^{-\epsilon n};$
$\displaystyle b_{mn}^{(1)}(\epsilon)$ $\displaystyle=\frac{16\pi
a_{n}^{(1)}(\epsilon)m}{4v(\epsilon)(ln(\epsilon+1)-ln1)\mu_{1}}=\frac{2e^{-\epsilon
n}\pi m}{(1-\frac{1}{\epsilon})(ln(\epsilon+1)-ln1)};$ $\displaystyle
y_{mn}^{(1)}(\epsilon)$
$\displaystyle=b_{mn}(\epsilon)\frac{l_{n}a_{n}^{(1)}(\epsilon)}{a_{n}^{(1)}(\epsilon)-1}.$

Then, it is easy to see that
$2\leq\sum_{n\in\mathbb{Z}}|\phi_{l}(x-n)|^{2}\leq 8$ for almost all
$x\in\mathbb{R}$. Hence, by Theorem 3.7, the collection
$\big{\\{}\big{|}\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,1)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}}$
is a frame with frame bounds $\frac{ln(\epsilon+1)}{\epsilon}$ and
$\frac{4ln(\epsilon+1)}{\epsilon}$.

Now, for any $f\in L^{2}(\mathbb{R},dx)$, we have

$\displaystyle\lim_{\epsilon\to 0^{+}}\sum_{m,n\in\mathbb{Z}}|\langle
f,\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,1)}\rangle|^{2}$
$\displaystyle\leq\lim_{\epsilon\to
0^{+}}\frac{4ln(\epsilon+1)}{\epsilon}\|f\|^{2}$ $\displaystyle\leq
4\|f\|^{2}$ $\displaystyle\leq\sum_{m,n\in\mathbb{Z}}|\langle
f,\phi^{1,0}_{(m,n,1)}\rangle|^{2}$ $\displaystyle\leq 16||f||^{2}.$

##  4\. A Perturbation Result

The importance of perturbation theory is well-known and has been extensively
studied, see [17]. For a variety of perturbation results for different types
of frames, we refer to [2, 9, 13]. In this section, we give Paley-Wiener type
perturbation results for multivariate frames for $L^{2}(\mathbb{R},dx)$. The
following result shows that frames for $L^{2}(\mathbb{R},dx)$ associated with
the Weyl-Heisenberg group are stable under small perturbation.

######  Theorem 4.1.

Let
$\big{\\{}\big{|}\phi^{P,Q}_{(m,n,l)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ be a frame for $L^{2}(\mathbb{R},dx)$ with frame
bounds $\alpha_{o}$ and $\beta_{o}$ and let
$\big{\\{}\big{|}\tilde{\phi}^{P,Q}_{(m,n,l)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}\subset L^{2}(\mathbb{R},dx)$. For non-negative real
number $\lambda$ and $M$ with $\lambda<\alpha_{o}$, assume that

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}-\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}$
$\displaystyle\leq M\min\Big{\\{}\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2},\
\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}\Big{\\}}+\lambda||f||^{2},$ (4.1)

for all $f\in L^{2}(\mathbb{R},dx)$. Then,
$\big{\\{}\big{|}\tilde{\phi}^{P,Q}_{(m,n,l)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ is a frame for $L^{2}(\mathbb{R},dx)$ with frame
bounds

$\displaystyle\frac{1}{2(M+1)}\Big{(}1-\frac{\lambda}{\alpha_{o}}\Big{)}\alpha_{o}\quad\text{and}\quad\Big{(}2\beta_{o}(M+1)+\lambda\Big{)}.$

###### Proof.

By hypothesis (4.1), for any $f\in L^{2}(\mathbb{R},dx)$, we have

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}$
$\displaystyle\leq\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}-\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}+2\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}$ $\displaystyle\leq
2M\min\Big{(}\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2},\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}\Big{)}+\lambda||f||^{2}$
$\displaystyle+2\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}$
$\displaystyle\leq(2M+2)\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2}+\lambda||f||^{2}$ $\displaystyle\leq
2(M+1)\beta_{o}||f||^{2}+\lambda||f||^{2}$
$\displaystyle=\big{(}2\beta_{o}(M+1)+\lambda\big{)}||f||^{2}.$ (4.2)

Similarly, using (4.1), for any $f\in L^{2}(\mathbb{R},dx)$, we can show that

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2}-\lambda||f||^{2}\leq(2M+2)\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2},$

which entails

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2}-\frac{\lambda}{\alpha_{o}}\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{P,Q}_{(m,n,l)}\rangle|^{2}\leq
2(M+1)\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}.$

That is

$\displaystyle\frac{1}{2(M+1)}\big{(}1-\frac{\lambda}{\alpha_{o}}\big{)}\alpha_{o}||f||^{2}\leq\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{P,Q}_{(m,n,l)}\rangle|^{2}\ \text{for all}\ f\in
L^{2}(\mathbb{R},dx).$ (4.3)

Hence, conclusion follows from (4) and (4.3). ∎

######  Remark 4.2.

Theorem 4.1 gives a perturbation result for multivariate Gabor frame such that
the new sequence
$\big{\\{}\big{|}\tilde{\phi}^{P,Q}_{(m,n,l)}\big{>}\big{\\}}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ is a Gabor frame. Using Theorem 3.8, this new Gabor
frame can be related to some wavelet frame
$\\{\big{|}\tilde{\phi}^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ associated with the extended affine group which can be
constructed using perturbation result. More precisely, Let
$\\{\big{|}\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ be a frame for $L^{2}(\mathbb{R},dx)$ with frame
bounds

$\displaystyle\frac{\alpha_{o}^{2}c}{\beta_{o}\epsilon}\big{(}\ln(\epsilon+c)-\ln
c\big{)}\ \text{and}\ \alpha_{o}c\big{(}\frac{\ln(\epsilon+c)-\ln
c}{\epsilon}\big{)},$

and let
$\\{\big{|}\tilde{\phi}^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}\subset L^{2}(\mathbb{R},dx)$. Let $\lambda$, $M\geq 1$
be such that
$\lambda<\frac{\alpha_{o}^{2}c}{\beta_{o}\epsilon}\big{(}\ln(\epsilon+c)-\ln
c\big{)}$. For any $f\in L^{2}(\mathbb{R},dx)$, assume that

$\displaystyle\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}-\tilde{\phi}^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\rangle|^{2}$
$\displaystyle\leq M\min\Big{\\{}\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\phi^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\rangle|^{2},\
\sum_{l=1}^{N}\sum_{m,n\in\mathbb{Z}}|\langle
f|\tilde{\phi}^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\rangle|^{2}\Big{\\}}+\lambda\|f\|^{2}.$

Then,
$\\{\big{|}\tilde{\phi}^{\epsilon,u(\epsilon),v(\epsilon)}_{(m,n,l)}\big{>}\\}_{m,n\in\mathbb{Z}\atop
l\in\\{1,2,\dots,N\\}}$ is a frame for $L^{2}(\mathbb{R},dx)$ with frame
bounds

$\displaystyle\frac{1}{2(M+1)}\Big{(}1-\frac{\lambda\beta\epsilon}{\alpha^{2}c(\ln(\epsilon+c)-\ln
c)}\Big{)}\Big{(}\frac{\alpha_{o}^{2}c}{\beta_{o}\epsilon}(\ln(\epsilon+c)-\ln
c)\Big{)}\ $ and $\displaystyle\
\big{(}\frac{2\alpha_{o}c}{\epsilon}(\ln(\epsilon+c)-\ln
c)(M+1)+\lambda\big{)}.$

##  5\. Conclusion

Gabor frames has potential applications in time-frequency analysis and signal
analysis. By using fundamental tools developed in [21] for the construction of
Parseval frames and using a unitary irreducible representation associated to
the Weyl-Heisenberg group, we give frame conditions for Gabor systems with
several generators in the signal space $L^{2}(\mathbb{R},dx)$. Afterwards, we
give frame conditions for wavelet systems with several generators associated
with an extended affine group. It would be interesting to know that: How
multivariate frames for the Weyl-Heisenberg group and the extended affine
group are related? We provide a relation between frame bounds of multivariate
Gabor systems and wavelet systems in the signal space $L^{2}(\mathbb{R},dx)$.
Stability of frames is equally important in study of stable signal analysis,
in particular, stable reconstruction and decomposition of signals. A stability
result, with explicit frame bounds, for Gabor frames of the signal space
$L^{2}(\mathbb{R},dx)$ is discussed. These results are useful in applications
of frames.

Data Related Statement

No data used in this study.

Conflicts of Interest

The authors have no conflicts of interest.

## References

  * [1] Ali, S. T., Antoine, J. P., Gazeau, J. P.: Coherent states, Wavelets and Their Generalizations. Springer-Verlag, New York (2000).
  * [2] Casazza, P. G., Kutyniok, G.: Finite frames: Theory and Applications. Birkh$\ddot{a}$user, New York (2012).
  * [3] Christensen, O.: An Introduction to Frames and Riesz Bases. Birkh$\ddot{a}$user, New York (2016).
  * [4] Claasen, T. A. C. M., Mecklenbrauker, W.F. G.: Wigner distribution–a tool for time-frequency signal analysis. Philips J. Res. 35, 217–250 (1980).
  * [5] Dahlke, S., Fornasier, M., Rauhut, H., Steidl, G., Teschke, G.: Generalized coorbit theory, Banach frames, and the relation to $\alpha$-modulation spaces. Proc. London Math. Soc. 96 (2) (2008), 464–506.
  * [6] Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992).
  * [7] Deepshikha, Vashisht, L. K.: Necessary and sufficient conditions for discrete wavelet frames in $\mathbb{C}^{N}$. J. Geom. Phys. 117, 134–143 (2017).
  * [8] Duffin R. J., Schaeffer, A. C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952).
  * [9] Favier, S. J., Zalik, R. A.: On the stability of frames and Riesz bases. Appl. Comp. Harmon. Anal. 2 (2), 160–173 (1995).
  * [10] Feichtinger, H. G., Strohmer, T.: Gabor analysis and algorithms, Birkhäuser, Boston (1998).
  * [11] Gabor, D.: Theory of communication. J. Inst. Elect. Eng. 93, 429–457 (1946).
  * [12] Gröchenig, K.: Foundations of Time-frequency Analysis, Birkhäuser Boston, Inc., Boston (2001).
  * [13] Heil, C.: A Basis Theory Primer (Expanded edition), Birkhäuser, New York (2011).
  * [14] Heil, C., Walnut, D. F.: Continuous and discrete wavelet transforms. SIAM Rev. 31 (4), 628–666 (1989).
  * [15] Jyoti, Vashisht, L. K.: $\mathcal{K}$-Matrix-valued wave packet frames in $L^{2}(\mathbb{R}^{d},\mathbb{C}^{s\times r})$. Math. Phys. Anal. Geom. 21 (3), Art 21, 19 pp (2018).
  * [16] Jyoti, Vashisht, L. K.: On matrix-valued wave packet frames in $L^{2}(\mathbb{R}^{d},\mathbb{C}^{s\times r})$. Anal. Math. Phys. 10 (4), Art 66, 24pp (2020).
  * [17] Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag Berlin Heidelberg (1976).
  * [18] Malhotra, H. K., Vashisht, L. K.: On vector-valued nonuniform multiresolution analysis. Appl. Anal. (2021), https://doi.org/10.1080/00036811.2021.1916479.
  * [19] Neumann, J. von: Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton. NJ, 1932/1945/1955.
  * [20] Perelomov, A.: Generalized Coherent States and Their Applications, Texts and Monographs in Physics. Springer-Verlag, Berlin (1986).
  * [21] Subag, E. M., Baruch, E. M., Birman J. L., Mann, A.: Gabor analysis as contraction of wavelets analysis. J. Math. Phys. 58 (8), Art No. 081702, 15 pp (2017).
  * [22] Torresani, B.: Wavelets associated with representations of the affine Weyl-Heisenberg group. J. Math. Phys. 32 (5), 1273–1279 (1991).
  * [23] Torresani, B.: Time-frequency representations: wavelet packets and optimal decomposition. Ann. Inst. H. Poincare Phys. Theor. 56 (2), 215–234 (1992).
  * [24] Vashisht, L. K., Deepshikha: Weaving properties of generalized continuous frames generated by an iterated function system. J. Geom. Phys. 110, 282–295 (2016).
  * [25] Zalik, R. A.: On MRA Riesz wavelets, Proc. Amer. Math. Soc. 135 (3), 787–793 (2007).
  * [26] Zalik, R. A.: Orthonormal wavelet systems and multiresolution analyses. J. Appl. Funct. Anal. 5 (1), 31–41 (2010).