# This is the title

GROUP-FRAMES FOR BANACH SPACES
K. MAHESH KRISHNA
Post Doctoral Fellow

Statistics and Mathematics Unit

Indian Statistical Institute, Bangalore Centre

Karnataka 560 059, India

Email<EMAIL_ADDRESS>
Date:

Abstract: In the literature, frames generated by unitary representations of
groups (known as group-frames) are studied only for Hilbert spaces. We make
first study of frames for Banach spaces generated by isometric invertible
representations of discrete groups on Banach spaces. These frames are
characterized using left regular, right regular, Gram-matrices and group-
matrices on classical sequence spaces. A sufficiently large collection of
functional-vector pairs using the double commutant of the representation is
identified which generate group-frames for Banach spaces. Subsequently, we
study Schauder frames generated by time-frequency shift operators on finite
dimensional Banach spaces. We derive Moyal formula, fundamental identity of
Gabor analysis, Wexler-Raz criterion and Ron-Shen duality in functional form.

Keywords: Frame, group, representation.

Mathematics Subject Classification (2020): 42C15, 43A65, 46B04, 46A45, 46B45.

###### Contents

  1. 1 Introduction
  2. 2 Unconditional Schauder frames generated by groups
  3. 3 Finite Gabor-Schauder frames
  4. 4 Acknowledgments

##  1\. Introduction

In their Memoirs ‘Frames, Bases and Group Representations’ [16], Han and
Larson initiated the study of Parseval frames for Hilbert spaces generated by
abstract countable groups which sheds light on the structure of unitary
representations of the given group. Recall that a countable group with
discrete topology is known as a discrete group. The definition of Parseval
frame reads as follows.

######  Definition 1.1.

[8, 16] Let $G$ be a countable set. A sequence $\\{\tau_{g}\\}_{g\in G}$ in a
Hilbert space $\mathcal{H}$ is said to be a Parseval frame for $\mathcal{H}$
if

$\displaystyle\|h\|^{2}=\sum_{g\in G}|\langle
h,\tau_{g}\rangle|^{2},\quad\forall h\in\mathcal{H}.$

Definition 1.1 is equivalent to the following equation:

(1) $\displaystyle h=\sum_{{g\in G}}\langle
h,\tau_{g}\rangle\tau_{g},\quad\forall h\in\mathcal{H}.$

One of the main objects in frame theory is to generate Parseval frames from a
given vector. In this regard, Han and Larson introduced the notion of frames
generated by groups as follows [16].

######  Definition 1.2.

[16] Let $G$ be a discrete group and $\\{\tau_{g}\\}_{g\in G}$ be a Parseval
frame for a Hilbert space $\mathcal{H}$. The frame $\\{\tau_{g}\\}_{g\in G}$
is said to be a group-frame if there exists a unitary representation $\pi$ of
$G$ on $\mathcal{H}$ and a vector $\tau\in\mathcal{H}$ such that

$\displaystyle\tau_{g}=\pi_{g}\tau,\quad\forall g\in G.$

In this case, the representation $\pi$ is called as frame representation and
$\tau$ is called as a frame vector.

After the introduction of group-frames, several important results appeared
which can be classified according to types of groups as follows.

  1. (1)

Finite groups [14, 31, 32, 33, 30, 10, 34, 15, 5, 22].

  2. (2)

Discrete groups [28, 29, 1, 2].

  3. (3)

locally compact groups [3, 7].

  4. (4)

compact groups [18].

  5. (5)

Lie groups [26, 25].

  6. (6)

ICC groups [9].

Similar to group-frames for Hilbert spaces, we can ask whether we can generate
frames for Banach spaces using groups. In this paper, we are interested in
frames for Banach spaces which are generated by discrete groups. Historically,
it was Grochenig [13] who introduced frames for Banach spaces known as Banach
frames which do not demand the reconstruction of element using series similar
to the one given in Equation (1). In 1999, Casazza, Han, and Larson [4]
introduced the notion of unconditional Schauder frames (also known as
framings) mainly taking the expansion property given in Equation (1). Let
$\mathcal{X}$ be a separable Banach space and $\mathcal{X}^{*}$ be its dual.

######  Definition 1.3.

[4] Let $G$ be a discrete group. Let $\\{\tau_{g}\\}_{g\in G}$ be a sequence
in $\mathcal{X}$ and $\\{f_{g}\\}_{g\in G}$ be a sequence in
$\mathcal{X}^{*}.$ The pair $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$
is said to be an unconditional Schauder frame (we write USF) for $\mathcal{X}$
if

(2) $\displaystyle x=\sum_{g\in G}f_{g}(x)\tau_{g},\quad\forall
x\in\mathcal{X},$

where the series in (2) converges unconditionally.

In this paper, we make a first attempt to understand USF which are generated
by representations of discrete groups in Banach spaces. We organize the paper
as follows. In Section 2 we start by mentioning the notion of group
representation in Banach spaces. Then we introduce the notion of group-frames
(Definition 2.2) and show that the notion reduces to Definition 1.2 for
Hilbert spaces with unitary representations. Then we introduce a subclass of
group-frames for Banach spaces called a group-p-unconditional Schauder frame
(group-p-USF) which factor through classical sequence spaces (Definition 2.3).

Theorem 2.8 shows that representations giving group-p-USFs can be obtained by
restricting the standard left regular representation on sequence spaces. A
connection between group-p-USFs and group-matrices is derived in Theorem 2.10.
We show in Theorem 2.11 group-p-USFs can be characterized using algebraic
equations. Theorem 2.13 gives a characterization of group-p-USFs using
standard left regular representation on sequence spaces. Proposition 2.14 and
Theorem 2.17 show that once a pair of functional and a vector which generate
group-p-USF is non-empty then the set of all such pairs is very large.

In Section 3 we do finite dimensional Gabor analysis in function form without
using the inner product. Even though it is true that finite dimensional Banach
spaces can be made a Hilbert space, rather putting an inner product, we
believe that it is best to study just by using functional and vectors. First
important result of this section is Banach space version of Moyal formula,
derived in Theorem 3.3. Using this result we get that we have a large supply
of Gabor-Schauder frames for finite dimensional Banach spaces (Corollary 3.4).
Then we study Gabor-Schauder frames generated by subgroups of the group of all
time-frequency shifts.

Theorem 3.8 derives fundamental identity of Gabor analysis for Banach spaces.
Wexler-Raz criterion for Gabor-Schauder frames is derived in Theorem 3.9. A
partial Ron-Shen duality for Gabor-Schauder frames is derived in Theorem 3.10.

##  2\. Unconditional Schauder frames generated by groups

Throughout $G$ denotes a group (need not be abelian). We denote the identity
element of $G$ by $e$. Given a Banach space $\mathcal{X}$,
$\mathcal{II}(\mathcal{X})$ be the set of all invertible linear isometries on
$\mathcal{X}$. The identity operator on $\mathcal{X}$ is denoted by
$I_{\mathcal{X}}$. We use the following definition of representation in Banach
spaces throughout.

######  Definition 2.1.

Let $\mathcal{X}$ be a Banach space and $G$ be a topological group. A map
$\pi:G\to\mathcal{II}(\mathcal{X})$ is said to be an invertible isometric
representation of $G$ if the following condition hold:

  1. (i)

$\pi:G\to\mathcal{II}(\mathcal{X})$ is a group homomorphism, i.e.,

$\displaystyle\pi_{g}\pi_{h}=\pi_{gh},\quad\forall g,h\in G.$

  2. (ii)

$\pi$ is continuous in the following sense. For each fixed $x\in\mathcal{X}$,
the map

$\displaystyle G\ni g\mapsto\pi_{g}x\in\mathcal{X}$

is continuous.

In this section, we consider only discrete groups. Hence the condition (ii) is
always satisfied. Given two invertible isometric representations
$\pi:G\to\mathcal{II}(\mathcal{X})$ and
$\Delta:G\to\mathcal{II}(\mathcal{Y})$, we say that they are equivalent if
there is an invertible operator $T:\mathcal{X}\to\mathcal{Y}$ which
intertwines $\pi$ and $\Delta$. If the intertwining operator is an invertible
isometry, then we say that representations are invertible isometric
equivalent. For any discrete group $G$ and $p\in[1,\infty)$, we note that $G$
always admit two invertible isometric representations on $\ell^{p}(G)$ defined
as follows. Let $\\{\delta_{g}\\}_{g\in G}$ be the standard Schauder basis for
$\ell^{p}(G)$. We denote the coordinate functionals associated to
$\\{\delta_{g}\\}_{g\in G}$ by $\\{\zeta_{g}\\}_{g\in G}$.

  1. (i)

$p$-left regular representation $\lambda:G\to\mathcal{II}(\ell^{p}(G))$
defined on $\\{\delta_{g}\\}_{g\in G}$ and extended linearly as

$\displaystyle\lambda_{g}\delta_{h}\coloneqq\delta_{gh},\quad\forall g,h\in
G.$

  2. (ii)

$p$-right regular representation $\rho:G\to\mathcal{II}(\ell^{p}(G))$ defined
on $\\{\delta_{g}\\}_{g\in G}$ and extended linearly as

$\displaystyle\rho_{g}\delta_{h}\coloneqq\delta_{hg^{-1}},\quad\forall g,h\in
G.$

With these preliminaries, we now set the following definition.

######  Definition 2.2.

Let $G$ be a discrete group and $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in
G})$ be an USF for a Banach space $\mathcal{X}$. The USF $(\\{f_{g}\\}_{g\in
G},\\{\tau_{g}\\}_{g\in G})$ is said to be a group-USF if there exist an
invertible isometric representation $\pi$ of $G$ on $\mathcal{X}$, a vector
$\tau\in\mathcal{X}$ and a functional $f\in\mathcal{X}^{*}$ such that

(3) $\displaystyle f_{g}=f\pi_{g^{-1}},\quad\tau_{g}=\pi_{g}\tau,\quad\forall
g\in G.$

We first show that Definition 2.2 truly generalizes Definition 1.2. Let
$\\{\tau_{g}\\}_{g\in G}=\\{\pi_{g}\tau\\}_{g\in G}$ be a group-frame for a
Hilbert space $\mathcal{H}$. Let $f$ be the functional on $\mathcal{H}$
defined by $\tau$, i.e., $f(h)=\langle h,\tau\rangle$, for all
$h\in\mathcal{H}$. Then

$\displaystyle
f_{g}(h)=f(\pi_{g^{-1}}h)=\langle\pi_{g^{-1}}h,\tau\rangle=\langle
h,\pi_{g}\tau\rangle=\langle h,\tau_{g}\rangle,\quad\forall h\in\mathcal{H}.$

Therefore $f_{g}$ is determined by $\tau_{g}$ for all $g\in G$.
It seems that we can not give a satisfactory theory for group-USF like that of
group-frames for Hilbert spaces. Therefore we study a class of group-USF’s
defined as follows. Our definition is motivated from the notion of
p-approximate Schauder frames defined in [21].

######  Definition 2.3.

Let $p\in[1,\infty)$. A group-USF $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in
G})$ for a Banach space $\mathcal{X}$ is said to be a group-p-USF if the
following conditions hold.

  1. (i)

The map (analysis operator) $\theta_{f}:\mathcal{X}\ni
x\mapsto\\{f_{g}(x)\\}_{g\in G}\in\ell^{p}(G)$ is a well-defined isometry.

  2. (ii)

The map (synthesis operator) $\theta_{\tau}:\ell^{p}(G)\ni\\{a_{g}\\}_{g\in
G}\mapsto\sum_{g\in G}a_{g}\tau_{g}\in\mathcal{X}$ is a well-defined bounded
linear operator.

In this case, the representation $\pi$ is called as p-USF representation and
the pair $(f,\tau)$ is called as a p-USF functional-vector.

We also need following generalization of Definition 2.3.

######  Definition 2.4.

Let $p\in[1,\infty)$ and $G$ be a discrete group. A USF $(\\{f_{g}\\}_{g\in
G},\\{\tau_{g}\\}_{g\in G})$ for a Banach space $\mathcal{X}$ is said to be a
p-USF if the following conditions hold.

  1. (i)

The map (analysis operator) $\theta_{f}:\mathcal{X}\ni
x\mapsto\\{f_{g}(x)\\}_{g\in G}\in\ell^{p}(G)$ is a well-defined isometry.

  2. (ii)

The map (synthesis operator) $\theta_{\tau}:\ell^{p}(G)\ni\\{a_{g}\\}_{g\in
G}\mapsto\sum_{g\in G}a_{g}\tau_{g}\in\mathcal{X}$ is a well-defined bounded
linear operator.

We begin by recording a characterization result of p-USFs which is motivated
from the characterization of Hilbert space frames by Holub [17].

######  Theorem 2.5.

A pair $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ is a p-USF for
$\mathcal{X}$, if and only if

$\displaystyle f_{g}=\zeta_{g}U,\quad\tau_{g}=V\delta_{g},\quad\forall g\in
G,$

where $U:\mathcal{X}\rightarrow\ell^{p}(G)$, $V:\ell^{p}(G)\to\mathcal{X}$ are
bounded linear operators such that $VU=I_{\mathcal{X}}$ and $U$ is an
isometry.

###### Proof.

$(\Leftarrow)$ Clearly $\theta_{f}$ and $\theta_{\tau}$ are bounded linear
operators. Now let $x\in\mathcal{X}$. Then

(4) $\displaystyle\sum_{{g\in G}}f_{g}(x)\tau_{g}=\sum_{{g\in
G}}\zeta_{g}(Ux)V\delta_{g}=V\left(\sum_{{g\in
G}}\zeta_{g}(Ux)\delta_{g}\right)=VUx=x.$

Note that, since $U$ is isometry,

$\displaystyle\|\theta_{f}x\|=\left\|\sum_{{g\in
G}}f_{g}(x)\delta_{g}\right\|=\left\|\sum_{{g\in
G}}\zeta_{g}(Ux)\delta_{g}\right\|=\|Ux\|=\|x\|,\quad\forall x\in\mathcal{X}.$

$(\Rightarrow)$ Define $U\coloneqq\theta_{f}$, $V\coloneqq\theta_{\tau}$. Then
$\zeta_{g}(Ux)=\zeta_{g}(\theta_{f}x)=\zeta_{g}(\\{f_{k}(x)\\}_{k\in
G})=f_{g}(x)$, $\forall x\in\mathcal{X}$,
$V\delta_{g}=\theta_{\tau}\delta_{g}=\tau_{g}$, $\forall g\in G$ and
$VU=\theta_{\tau}\theta_{f}=I_{\mathcal{X}}$. Since $\theta_{f}$ is an
isometry, we also have $U$ is an isometry. ∎

We record the following important result from [21].

######  Theorem 2.6.

[21] Let $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ be a p-USF for
$\mathcal{X}$. Then

  1. (i)

$I_{\mathcal{X}}=\theta_{\tau}\theta_{f}.$

  2. (ii)

$G_{f,\tau}\coloneqq\theta_{f}\theta_{\tau}:\ell^{p}(G)\to\ell^{p}(G)$ is a
projection onto $\theta_{f}(\mathcal{X})$.

In [16], Han and Larson showed that upto unitary operator, frame
representation is a piece of left regular representation. With the help of
following lemma we generalize their result for Banach spaces.

######  Lemma 2.7.

If $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ is a group-p-USF for
$\mathcal{X}$, then range of its analysis operator is invariant under p-left-
regular representation of $G$, i.e.,
$\lambda_{g}(\theta_{f}(\mathcal{X}))\subseteq\theta_{f}(\mathcal{X})$ for all
$g\in G$.

###### Proof.

For any $g\in G$,

$\displaystyle\lambda_{g}\theta_{f}x$
$\displaystyle=\lambda_{g}\left(\sum_{h\in
G}f_{h}(x)\delta_{h}\right)=\sum_{h\in G}f_{h}(x)\delta_{gh}=\sum_{u\in
G}f_{g^{-1}u}(x)\delta_{u}$ $\displaystyle=\sum_{u\in
G}f(\pi_{u^{-1}g}x)\delta_{u}=\sum_{u\in
G}(f\pi_{u^{-1}})(\pi_{g}x)\delta_{u}=\theta_{f}(\pi_{g}x)\in\theta_{f}(\mathcal{X}),\quad\forall
x\in\mathcal{X}.$

∎

######  Theorem 2.8.

Every group-p-USF representation $\pi$ of $G$ is invertibly isometrically
equivalent to a subrepresentation of p-left-regular representation $\lambda$
of $G$.

###### Proof.

Let $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ be a group-p-USF for
$\mathcal{X}$. Lemma 2.7 says that
$\lambda_{g}(\theta_{f}(\mathcal{X}))\subseteq\theta_{f}(\mathcal{X})$ for all
$g\in G$. Therefore the map

$\displaystyle\Delta_{g}\coloneqq\lambda_{g}|_{\theta_{f}(\mathcal{X})}:\theta_{f}(\mathcal{X})\to\theta_{f}(\mathcal{X}),\quad\forall
g\in G$

is a well-defined invertible isometric representation of $G$. We show that
$\Delta$ and $\pi$ are isometrically invertibly equivalent. Note that
$\theta_{f}:\mathcal{X}\to\theta_{f}(\mathcal{X})$ is an invertible isometry.
We are done if we show that $\theta_{f}:\mathcal{X}\to\theta_{f}(\mathcal{X})$
intertwines $\Delta$ and $\pi$. This follows by doing a similar calculation as
in the proof of Lemma 2.7. ∎

Vale and Waldron discovered that for finite groups, groups-frames can be
characterized using group-matrices [30]. We show that their result remains
valid for Banach spaces. First we recall the definition of group-matrix.

######  Definition 2.9.

[19] Let $G$ be a discrete group. A matrix $A\coloneqq[a_{g,h}]_{g,h\in G}$
over $\mathbb{C}$ is said to be a group-matrix if there exists a function
$\nu:G\to\mathbb{C}$ such that

(5) $\displaystyle a_{g,h}=\nu(g^{-1}h),\quad\forall g,h\in G.$

Let $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ be a group-p-USF for
$\mathcal{X}$. Then we note that

$\displaystyle
f_{g}(\tau_{h})=(f\pi_{g^{-1}})(\pi_{h}\tau)=f(\pi_{g^{-1}h}\tau),\quad\forall
g,h\in G.$

Let $G_{f,\tau}\coloneqq\theta_{f}\theta_{\tau}$ be the Gramian of
$(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ whose matrix w.r.t. the
standard Schauder basis $\\{\delta_{g}\\}_{g\in G}$ for $\ell^{p}(G)$ is given
by

$\displaystyle[f_{g}(\tau_{h})]_{g,h\in G}.$

Now by defining $\nu:G\ni g\mapsto\nu(g)\coloneqq
f(\pi_{g}\tau)\in\mathbb{C}$, we see from Equation (5) that $G_{f,\tau}$ is a
group-matrix. Next theorem shows that converse of this also holds.

######  Theorem 2.10.

Let $G$ be a discrete group. Then a p-USF $(\\{f_{g}\\}_{g\in
G},\\{\tau_{g}\\}_{g\in G})$ for $\mathcal{X}$ is a group-p-USF for
$\mathcal{X}$ if and only if its Gramian $G_{f,\tau}$ is a group-matrix.

###### Proof.

As we already derived only if part, we prove if part. Assume that the Gramian
$G_{f,\tau}$ is a group-matrix. Then there exists a function
$\nu:G\to\mathbb{C}$ such that

$\displaystyle\nu(g^{-1}h)=f_{g}(\tau_{h}),\quad\forall g,h\in G.$

Given $g\in G$, define

$\displaystyle\pi_{g}:\mathcal{X}\ni x\mapsto\pi_{g}x\coloneqq\sum_{h\in
G}f_{h}(x)\tau_{gh}\in\mathcal{X}$

and

$\displaystyle\Delta_{g}:\mathcal{X}\ni x\mapsto\Delta_{g}x\coloneqq\sum_{h\in
G}f_{gh}(x)\tau_{h}\in\mathcal{X}.$

Note that

(6) $\displaystyle
f_{gu}(\tau_{gh})=\nu((gu)^{-1}gh)=\nu(u^{-1}h)=f_{h}(\tau_{u}),\quad\forall
u,g,h\in G.$

Using Equation (6), we get

$\displaystyle\Delta_{g}\pi_{g}x$ $\displaystyle=\Delta_{g}\left(\sum_{h\in
G}f_{h}(x)\tau_{gh}\right)=\sum_{h\in G}f_{h}(x)\Delta_{g}\tau_{gh}=\sum_{h\in
G}f_{h}(x)\sum_{u\in G}f_{gu}(\tau_{gh})\tau_{u}$ $\displaystyle=\sum_{h\in
G}f_{h}(x)\sum_{u\in G}f_{u}(\tau_{h})\tau_{u}=\sum_{h\in
G}f_{h}(x)\tau_{h}=x,\quad\forall x\in\mathcal{X}$

and

$\displaystyle\pi_{g}\Delta_{g}x$ $\displaystyle=\pi_{g}\left(\sum_{h\in
G}f_{gh}(x)\tau_{h}\right)=\sum_{h\in G}f_{gh}(x)\pi_{g}\tau_{h}=\sum_{h\in
G}f_{gh}(x)\sum_{u\in G}f_{u}(\tau_{h})\tau_{gu}$ $\displaystyle=\sum_{h\in
G}f_{gh}(x)\sum_{u\in G}f_{gu}(\tau_{gh})\tau_{gu}=\sum_{h\in
G}f_{gh}(x)\tau_{gh}=x,\quad\forall x\in\mathcal{X}.$

Therefore $\Delta_{g}$ is the inverse of $\pi_{g}$. We next show $\pi_{g}$ is
isometry. For $x\in\mathcal{X}$, using Equation (6) and $\theta_{f}$ is an
isometry,

$\displaystyle\|\pi_{g}x\|^{p}$
$\displaystyle=\|\theta_{f}(\pi_{g}x)\|^{p}=\sum_{h\in
G}|f_{h}(\pi_{g}x)|^{p}=\sum_{h\in G}\left|f_{h}\left(\sum_{u\in
G}f_{u}(x)\tau_{gu}\right)\right|^{p}$ $\displaystyle=\sum_{h\in
G}\left|\sum_{u\in G}f_{u}(x)f_{h}(\tau_{gu})\right|^{p}=\sum_{h\in
G}\left|\sum_{u\in G}f_{u}(x)f_{gg^{-1}h}(\tau_{gu})\right|^{p}$
$\displaystyle=\sum_{h\in G}\left|\sum_{u\in
G}f_{u}(x)f_{g^{-1}h}(\tau_{u})\right|^{p}=\sum_{h\in
G}\left|f_{g^{-1}h}\left(\sum_{u\in G}f_{u}(x)\tau_{u}\right)\right|^{p}$
$\displaystyle=\sum_{h\in G}|f_{g^{-1}h}(x)|^{p}=\|x\|^{p}.$

To show Equation (3) again using Equation (6),

$\displaystyle\pi_{g}\tau_{e}$ $\displaystyle=\sum_{h\in
G}f_{h}(\tau_{e})\tau_{gh}=\sum_{u\in
G}f_{g^{-1}u}(\tau_{e})\tau_{u}=\sum_{u\in
G}f_{g^{-1}u}(\tau_{g^{-1}g})\tau_{u}$ $\displaystyle=\sum_{u\in
G}f_{u}(\tau_{g})\tau_{u}=\tau_{g},\quad\forall g\in G$

and

$\displaystyle(f_{e}\pi_{g^{-1}})(x)$ $\displaystyle=f_{e}\left(\sum_{h\in
G}f_{h}(x)\tau_{g^{-1}h}\right)=\sum_{h\in
G}f_{h}(x)f_{e}(\tau_{g^{-1}h})=\sum_{h\in
G}f_{h}(x)f_{g^{-1}g}(\tau_{g^{-1}h})$ $\displaystyle=\sum_{h\in
G}f_{h}(x)f_{g}(\tau_{h})=f_{g}\left(\sum_{h\in
G}f_{h}(x)\tau_{h}\right)=f_{g}(x),\quad\forall g\in G,\forall
x\in\mathcal{X}.$

We are left with showing that $\pi$ is a homomorphism. Let $g,h\in G$ and
$x\in\mathcal{X}$. Then using Equation (6),

$\displaystyle\pi_{g}\pi_{h}x$ $\displaystyle=\sum_{u\in
G}f_{u}(\pi_{h}x)\tau_{gu}=\sum_{u\in G}f_{u}\left(\sum_{v\in
G}f_{v}(x)\tau_{hv}\right)\tau_{gu}$ $\displaystyle=\sum_{u\in G}\sum_{v\in
G}f_{v}(x)f_{u}(\tau_{hv})\tau_{gu}=\sum_{u\in G}\sum_{v\in
G}f_{v}(x)f_{hh^{-1}u}(\tau_{hv})\tau_{gu}$ $\displaystyle=\sum_{u\in
G}\sum_{v\in G}f_{v}(x)f_{h^{-1}u}(\tau_{v})\tau_{gu}=\sum_{u\in
G}f_{h^{-1}u}\left(\sum_{v\in G}f_{v}(x)\tau_{v}\right)\tau_{gu}$
$\displaystyle=\sum_{u\in G}f_{h^{-1}u}(x)\tau_{gu}=\sum_{v\in
G}f_{v}(x)\tau_{(gh)v}=\pi_{gh}x.$

∎

Kaftal, Larson and Zhang showed that group-frames can be characterized by
using an algebraic equation and involving inner proved (actually they proved
it in the setup of operator-valued frames) [20]. In the following result we
generalize the result of Kaftal, Larson and Zhang to Banach spaces.

######  Theorem 2.11.

Let $G$ be a discrete group and $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in
G})$ be a p-USF for $\mathcal{X}$. Then there is an invertible isometric
representation $\pi$ of $G$ on $\mathcal{X}$ for which

(7) $\displaystyle\tau_{g}=\pi_{g}\tau_{e},\quad
f_{g}=f_{e}\pi_{g^{-1}},\quad\forall g\in G$

if and only if

(8) $\displaystyle f_{ug}(\tau_{uh})=f_{g}(\tau_{h}),\quad\forall u,g,h\in G.$

Moreover, the representation can be defined as

(9)
$\displaystyle\pi_{g}\coloneqq\theta_{\tau}\lambda_{g}\theta_{f},\quad\forall
g\in G.$

###### Proof.

$(\Rightarrow)$ Let $u,g,h\in G$. Then we have

$\displaystyle f_{ug}(\tau_{uh})$
$\displaystyle=(f_{e}\pi_{(ug)^{-1}})(\pi_{uh}\tau_{e})=(f_{e}\pi_{g^{-1}u^{-1}})(\pi_{uh}\tau_{e})=(f_{e}\pi_{g^{-1}})(\pi_{h}\tau_{e})=f_{g}(\tau_{h}).$

$(\Leftarrow)$ Let $\pi_{g}$ be defined by Equation (9). We are required to
show that $\pi$ is an invertible isometric representation and satisfies
Equation (7). We first show that it satisfies Equation (7). To do so, we claim
the identity

(10)
$\displaystyle\lambda_{g}\theta_{f}\theta_{\tau}=\theta_{f}\theta_{\tau}\lambda_{g},\quad\forall
g\in G.$

Using Equation (8) we verify Equation (10). Note that it suffices to verify
Equation (10) at the standard basis vectors $\delta_{h}$, $h\in G$. Consider

$\displaystyle\theta_{f}\theta_{\tau}\lambda_{g}\delta_{h}$
$\displaystyle=\theta_{f}\theta_{\tau}\delta_{gh}=\theta_{f}\tau_{gh}=\sum_{u\in
G}f_{u}(\tau_{gh})\delta_{u}=\sum_{u\in
G}f_{gu}(\tau_{gh})\delta_{gu}=\sum_{u\in G}f_{u}(\tau_{h})\delta_{gu}$
$\displaystyle=\sum_{u\in
G}f_{u}(\tau_{gh})\lambda_{g}\delta_{u}=\lambda_{g}\left(\sum_{u\in
G}f_{u}(\tau_{h})\delta_{u}\right)=\lambda_{g}\theta_{f}\tau_{h}=\lambda_{g}\theta_{f}\theta_{\tau}\delta_{h}.$

Now using Equation (10)

$\displaystyle\pi_{g}\tau_{e}=\theta_{\tau}\lambda_{g}\theta_{f}\tau_{e}=\theta_{\tau}\lambda_{g}\theta_{f}\theta_{\tau}\delta_{e}=\theta_{\tau}\theta_{f}\theta_{\tau}\lambda_{g}\delta_{e}=I_{\mathcal{X}}\theta_{\tau}\lambda_{g}\delta_{e}=\theta_{\tau}\delta_{g}=\tau_{g},\quad\forall
g\in G$

and

$\displaystyle f_{e}(\pi_{g^{-1}}x)$
$\displaystyle=f_{e}(\theta_{\tau}\lambda_{g^{-1}}\theta_{f}x)=f_{e}\left(\sum_{u\in
G}f_{u}(x)\theta_{\tau}\lambda_{g^{-1}}\delta_{u}\right)=f_{e}\left(\sum_{u\in
G}f_{u}(x)\tau_{g^{-1}u}\right)$ $\displaystyle=\sum_{u\in
G}f_{u}(x)f_{e}(\tau_{g^{-1}u})=\sum_{u\in
G}f_{u}(x)f_{g^{-1}g}(\tau_{g^{-1}u})=\sum_{u\in G}f_{u}(x)f_{g}(\tau_{u})$
$\displaystyle=f_{g}\left(\sum_{u\in
G}f_{u}(x)\tau_{u}\right)=f_{g}(x),\quad\forall x\in\mathcal{X},\forall g\in
G.$

Now we show that $\pi$ is an invertible isometric representation. First we
need to show that it is bijective. We note that, for $g\in G$, the operator
$\Delta_{g}\coloneqq\theta_{\tau}\lambda_{g^{-1}}\theta_{f}$ is the inverse of
$\pi_{g}$. In fact, using Equation (10),

$\displaystyle\pi_{g}\Delta_{g}=\theta_{\tau}\lambda_{g}\theta_{f}\theta_{\tau}\lambda_{g^{-1}}\theta_{f}=\theta_{\tau}\theta_{f}\theta_{\tau}\lambda_{g}\lambda_{g^{-1}}\theta_{f}=I_{\mathcal{X}},$
$\displaystyle\Delta_{g}\pi_{g}=\theta_{\tau}\lambda_{g^{-1}}\theta_{f}\theta_{\tau}\lambda_{g}\theta_{f}=\theta_{\tau}\lambda_{g^{-1}}\lambda_{g}\theta_{f}\theta_{\tau}\theta_{f}=I_{\mathcal{X}}.$

To show $\pi_{g}$ is an isometry, given $x\in\mathcal{X}$,

$\displaystyle\|\pi_{g}x\|^{p}$ $\displaystyle=\sum_{h\in
G}|f_{h}(\pi_{g}x)|^{p}=\sum_{h\in
G}|f_{h}(\theta_{\tau}\lambda_{g}\theta_{f}x)|^{p}=\sum_{h\in
G}\left|(f_{h}\theta_{\tau}\lambda_{g})\left(\sum_{u\in
G}f_{u}(x)\delta_{u}\right)\right|^{p}$ $\displaystyle=\sum_{h\in
G}\left|f_{h}\left(\sum_{u\in
G}f_{u}(x)\theta_{\tau}\lambda_{g}\delta_{u}\right)\right|^{p}=\sum_{h\in
G}\left|f_{h}\left(\sum_{u\in G}f_{u}(x)\tau_{gu}\right)\right|^{p}$
$\displaystyle=\sum_{h\in G}\left|\sum_{u\in
G}f_{u}(x)f_{h}(\tau_{gu})\right|^{p}=\sum_{h\in G}\left|\sum_{u\in
G}f_{u}(x)f_{gg^{-1}h}(\tau_{gu})\right|^{p}$ $\displaystyle=\sum_{h\in
G}\left|\sum_{u\in G}f_{u}(x)f_{g^{-1}h}(\tau_{u})\right|^{p}=\sum_{h\in
G}\left|f_{g^{-1}h}\left(\sum_{u\in G}f_{u}(x)\tau_{u}\right)\right|^{p}$
$\displaystyle=\sum_{h\in G}|f_{g^{-1}h}(x)|^{p}=\|x\|^{p}.$

Finally, we show that $\pi$ is representation:

$\displaystyle\pi_{g}\pi_{h}$
$\displaystyle=\theta_{\tau}\lambda_{g}\theta_{f}\theta_{\tau}\lambda_{h}\theta_{f}=\theta_{\tau}\theta_{f}\theta_{\tau}\lambda_{g}\lambda_{h}\theta_{f}=I_{\mathcal{X}}\theta_{\tau}\lambda_{gh}\theta_{f}=\pi_{gh},\quad\forall
g,h\in G.$

∎

A careful observation on proof of “if” part of Theorem 2.11 gives the
following result.

######  Theorem 2.12.

Let $G$ be a discrete group and $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in
G})$ be a group-p-USF for $\mathcal{X}$. Then
$\lambda_{g}\theta_{f}\theta_{\tau}=\theta_{f}\theta_{\tau}\lambda_{g},$
$\forall g\in G.$

Next we try to relate frame representation with p-right regular
representation. This result was first derived by Mendez, Bodmann, Baker,
Bullock and McLaney in the context of binary frame [23].

######  Theorem 2.13.

Let $G$ be a discrete group and $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in
G})$ be a p-USF for $\mathcal{X}$. Then there is an invertible isometric
representation $\pi$ of $G$ on $\mathcal{X}$ for which

$\displaystyle\tau_{g}=\pi_{g}\tau_{e},\quad
f_{g}=f_{e}\pi_{g^{-1}},\quad\forall g\in G$

if and only if

$\displaystyle\theta_{f}\theta_{\tau}\\{a_{h}\\}_{h\in G}=\sum_{g\in
G}\eta(g)\rho_{g}\\{a_{h}\\}_{h\in G},\quad\forall\\{a_{h}\\}_{h\in
G}\in\ell^{p}(G),$

where

(11) $\displaystyle\eta:G\ni g\mapsto\eta(g)\coloneqq
f_{e}(\pi_{g}\tau_{e})\in\mathbb{C}.$

Moreover, the invertible isometric representation can be defined as

(12)
$\displaystyle\pi_{g}\coloneqq\theta_{\tau}\lambda_{g}\theta_{f},\quad\forall
g\in G.$

###### Proof.

$(\Rightarrow)$ Let $\eta$ be the function defined in Equation (11). Now for
each $\delta_{h}$,

$\displaystyle\theta_{f}\theta_{\tau}\delta_{h}$
$\displaystyle=\theta_{f}\tau_{h}=\sum_{u\in
G}f_{u}(\tau_{h})\delta_{u}=\sum_{u\in
G}(f\pi_{u^{-1}})(\pi_{h}\tau)\delta_{u}$ $\displaystyle=\sum_{u\in
G}f(\pi_{u^{-1}h}\tau)\delta_{u}=\sum_{g\in
G}f(\pi_{g}\tau)\delta_{hg^{-1}}=\sum_{g\in
G}f(\pi_{g}\tau)\rho_{g}\delta_{h}$ $\displaystyle=\left(\sum_{g\in
G}f(\pi_{g}\tau)\rho_{g}\right)\delta_{h}=\left(\sum_{g\in
G}\eta(g)\rho_{g}\right)\delta_{h}.$

$(\Leftarrow)$ We note that p-left and p-right regular representations
commute. In fact, for any $g,h\in G$ and for each standard basis vector
$\delta_{u}$, we have

$\displaystyle\lambda_{g}\rho_{h}\delta_{u}=\lambda_{g}\delta_{uh^{-1}}=\delta_{guh^{-1}}=\rho_{h}\delta_{gu}=\rho_{h}\lambda_{g}\delta_{u}.$

We then get

$\displaystyle\lambda_{g}\theta_{f}\theta_{\tau}\delta_{h}$
$\displaystyle=\lambda_{g}\left(\sum_{u\in
G}\eta(u)\rho_{u}\delta_{h}\right)=\sum_{u\in
G}\eta(u)\lambda_{g}\rho_{u}\delta_{h}$ $\displaystyle=\left(\sum_{u\in
G}\eta(u)\rho_{u}\right)\lambda_{g}\delta_{h}=\theta_{f}\theta_{\tau}\lambda_{g}\delta_{h},\quad\forall
g\in G.$

Now we claim the following:

$\displaystyle f_{ug}(\tau_{uh})=f_{g}(\tau_{h}),\quad\forall u,g,h\in G.$

Consider

$\displaystyle f_{ug}(\tau_{uh})$
$\displaystyle=f_{ug}(\theta_{\tau}\delta_{uh})=\zeta_{ug}(\theta_{f}\theta_{\tau}\delta_{uh})=\zeta_{ug}\left(\sum_{v\in
G}\eta(v)\rho_{v}\delta_{uh}\right)$ $\displaystyle=\sum_{v\in
G}\eta(v)\zeta_{ug}\rho_{v}\delta_{uh}=\sum_{v\in
G}\eta(v)\zeta_{ug}\delta_{uhv^{-1}}=\eta(g^{-1}h)=f_{g}(\tau_{h}),\quad\forall
u,g,h\in G.$

Hence claim holds. For each $g\in G$, we now define $\pi_{g}$ as in Equation
(12). Now by doing a similar calculation as in the converse part of proof of
Theorem 2.11 we get that $(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ is a
group-p-USF for $\mathcal{X}$. ∎

After giving several characterizations for frame representations, we next seek
to determine the collection of functionals and vectors which generate group-
frames. In the case of Hilbert spaces, Han and Larson completely characterized
vectors which generate group-frames using double commutant of image of
representation [16]. Even though we are unable to achieve this, we show that
certain large sets generate group-frames for Banach spaces. We first set some
notations. Let $G$ be a discrete group and $\pi:G\to\mathcal{II}(\mathcal{X})$
be a invertible isometric representation. Assume that there is a vector
$\tau\in\mathcal{X}$ and a functional $f\in\mathcal{X}^{*}$ such that
$(\\{f_{g}\\}_{g\in G},\\{\tau_{g}\\}_{g\in G})$ is a group-p-USF for
$\mathcal{X}$. Define the set of all group-p-USF vectors as

$\displaystyle\mathscr{F}_{G}(\pi,\mathcal{X})\coloneqq\\{(f_{1},\tau_{1})\in\mathcal{X}^{*}\times\mathcal{X}:(\\{{f_{1}}_{g}\\}_{g\in
G},\\{{\tau_{1}}_{g}\\}_{g\in G})\text{ is a group-p-USF for }\mathcal{X}\\}.$

By assumption $\mathscr{F}_{G}(\pi,\mathcal{X})\neq\emptyset$. We naturally
ask what is the structure of $\mathscr{F}_{G}(\pi,\mathcal{X})$? In the
following proposition we show that this set is quite large. We use the
following notation in sequel. Given a subset $\mathcal{A}$ of linear operators
on $\mathcal{X}$, by $\mathbb{I}^{*}(\mathcal{A})$ we mean the set of
invertible isometric operators $U:\mathcal{X}\to\mathcal{X}$ such that
$U\in\mathcal{A}$. Given $f\in\mathcal{X}^{*}$ and $\tau\in\mathcal{X}$, we
define

$\displaystyle
f[\mathbb{I}^{*}(\mathcal{A})^{-1},\mathbb{I}^{*}(\mathcal{A})]\tau\coloneqq\\{(fU^{-1},U\tau):U\in\mathbb{I}^{*}(\mathcal{A})\\}.$

######  Proposition 2.14.

If an invertible isometric representation $\pi:G\to\mathcal{II}(\mathcal{X})$
admits a functional-frame vector $(f,\tau)$, then

(13) $\displaystyle
f[\mathbb{I}^{*}(\pi(G)^{\prime})^{-1},\mathbb{I}^{*}(\pi(G)^{\prime})]\tau\subseteq\mathscr{F}_{G}(\pi,\mathcal{X}).$

###### Proof.

Let $(f_{1},\tau_{1})\in
f[\mathbb{I}^{*}(\pi(G)^{\prime})^{-1},\mathbb{I}^{*}(\pi(G)^{\prime})]\tau$.
Then $f_{1}=fU^{-1}$, $\tau_{1}=U\tau$ for some invertible isometry
$U:\mathcal{X}\to\mathcal{X}$ such that $U\in\mathbb{I}^{*}(\pi(G)^{\prime})$.
Define ${f_{2}}_{g}\coloneqq f_{1}\pi_{g^{-1}}$ and
${\tau_{2}}_{g}\coloneqq\pi_{g}\tau_{1}$, for all $g\in G$. Now we see that

$\displaystyle\sum_{g\in G}{f_{2}}_{g}(x){\tau_{2}}_{g}$
$\displaystyle=\sum_{g\in G}(f_{1}\pi_{g^{-1}})(x)\pi_{g}\tau_{1}=\sum_{g\in
G}(fU^{-1})(\pi_{g^{-1}}x)\pi_{g}U\tau$ $\displaystyle=\sum_{g\in
G}(f\pi_{g^{-1}})(U^{-1}x)U\pi_{g}\tau=U\left(\sum_{g\in
G}(f\pi_{g^{-1}})(U^{-1}x)\pi_{g}\tau\right)$ $\displaystyle=U\left(\sum_{g\in
G}f_{g}(U^{-1}x)\tau_{g}\right)=UU^{-1}x=x,\quad\forall x\in\mathcal{X}.$

Therefore $(\\{{f_{2}}_{g}\\}_{g\in G},\\{{\tau_{2}}_{g}\\}_{g\in G})$ is a
group-p-USF for $\mathcal{X}$ and consequently
$(f_{1},\tau_{1})\in\mathscr{F}_{G}(\pi,\mathcal{X})$. ∎

We next try to show $\mathscr{F}_{G}(\pi,\mathcal{X})$ has another large set
inside it. For this, we need following two results.

######  Theorem 2.15.

For any group $G$, and any $1\leq p<\infty$,

$\displaystyle\lambda(G)^{\prime}=\rho(G)^{\prime\prime}\quad\text{ and
}\quad\rho(G)^{\prime}=\lambda(G)^{\prime\prime}.$

###### Proof.

We already know from the proof of Theorem 2.13 that p-left and p-right regular
representations commute, i.e.,

$\displaystyle\lambda_{g}\rho_{h}=\rho_{h}\lambda_{g},\quad\forall g,h\in G.$

Hence for every $g\in G$, $\lambda_{g}\in\rho(G)^{\prime}$. By varying $g$, we
get $\lambda(G)\subseteq\rho(G)^{\prime}$. Taking commutant yield
$\lambda(G)^{\prime}\supseteq\rho(G)^{\prime\prime}$. Now we prove the reverse
inclusion. Let $T\in\lambda(G)^{\prime}$. To show $T\in\rho(G)^{\prime\prime}$
we need to show that $TS=ST$ for all $S\in\rho(G)^{\prime}$. So let
$S\in\rho(G)^{\prime}$. Note that to verify $TS=ST$, it suffices to verify
$TS\delta_{h}=ST\delta_{h}$ for all $h\in G$. Let $h\in G$. Then

$\displaystyle TS\delta_{h}$
$\displaystyle=TS\rho_{h^{-1}}\delta_{e}=T\rho_{h^{-1}}S\delta_{e}=T\rho_{h^{-1}}\left(\sum_{g\in
G}\zeta_{g}(S\delta_{e})\delta_{g}\right)$ $\displaystyle=\sum_{g\in
G}\zeta_{g}(S\delta_{e})T\rho_{h^{-1}}\delta_{g}=\sum_{g\in
G}\zeta_{g}(S\delta_{e})T\delta_{gh}=\sum_{g\in
G}\zeta_{g}(S\delta_{e})T\lambda_{gh}\delta_{e}$ $\displaystyle=\sum_{g\in
G}\zeta_{g}(S\delta_{e})\lambda_{gh}T\delta_{e}=\sum_{g\in
G}\zeta_{g}(S\delta_{e})\lambda_{gh}\left(\sum_{u\in
G}\zeta_{u}(T\delta_{e})\delta_{u}\right)$ $\displaystyle=\sum_{g\in
G}\zeta_{g}(S\delta_{e})\sum_{u\in
G}\zeta_{u}(T\delta_{e})\lambda_{gh}\delta_{u}=\sum_{g\in G}\sum_{u\in
G}\zeta_{g}(S\delta_{e})\zeta_{u}(T\delta_{e})\delta_{ghu}$

and

$\displaystyle ST\delta_{h}$
$\displaystyle=ST\lambda_{h}\delta_{e}=S\lambda_{h}T\delta_{e}=S\lambda_{h}\left(\sum_{u\in
G}\zeta_{u}(T\delta_{e})\delta_{u}\right)$ $\displaystyle=\sum_{u\in
G}\zeta_{u}(T\delta_{e})S\lambda_{h}\delta_{u}=\sum_{u\in
G}\zeta_{u}(T\delta_{e})S\delta_{hu}=\sum_{u\in
G}\zeta_{u}(T\delta_{e})S\rho_{(hu)^{-1}}\delta_{e}$ $\displaystyle=\sum_{u\in
G}\zeta_{u}(T\delta_{e})\rho_{(hu)^{-1}}S\delta_{e}=\sum_{u\in
G}\zeta_{u}(T\delta_{e})\rho_{(hu)^{-1}}\left(\sum_{g\in
G}\zeta_{g}(S\delta_{e})\delta_{g}\right)$ $\displaystyle=\sum_{u\in
G}\zeta_{u}(T\delta_{e})\sum_{g\in
G}\zeta_{g}(S\delta_{e})\rho_{(hu)^{-1}}\delta_{g}=\sum_{u\in G}\sum_{g\in
G}\zeta_{u}(T\delta_{e})\zeta_{g}(S\delta_{e})\delta_{ghu}.$

Therefore $\lambda(G)^{\prime}=\rho(G)^{\prime\prime}$. Finally
$\lambda(G)^{\prime\prime}=\rho(G)^{\prime\prime\prime}=\rho(G)^{\prime}$. ∎

######  Theorem 2.16.

For any discrete group $G$ and for every $1\leq p<\infty$, the map

$\displaystyle\Phi:\rho(G)^{\prime\prime}\ni A\mapsto\Phi(A)\coloneqq
JAJ\in\lambda(G)^{\prime\prime}$

is an algebra isomorphism, where

$\displaystyle J:\ell^{p}(G)\ni\\{a_{g}\\}_{g\in G}\mapsto J\\{a_{g}\\}_{g\in
G}\coloneqq\sum_{g\in G}a_{g}\delta_{g^{-1}}\in\ell^{p}(G).$

Moreover, we have the following.

  1. (i)

If $U\in\rho(G)^{\prime\prime}$ is invertible (resp. isometry), then $\Phi(U)$
is invertible (resp. isometry).

  2. (ii)

If $V\in\lambda(G)^{\prime\prime}$ is invertible (resp. isometry), then
$\Phi^{-1}(V)$ is invertible (resp. isometry).

###### Proof.

We first note that $J$ is an isomorphism and $J^{2}=I_{\ell^{p}(G)}$. Before
showing $\Phi$ is an isomorphism, we need to show that it is well-defined. Let
$A\in\rho(G)^{\prime\prime}$. We try to show that
$JAJ\in\lambda(G)^{\prime\prime}$ which says $\Phi$ is well-defined. By
Theorem 2.15, showing $JAJ\in\lambda(G)^{\prime\prime}$ is same as showing
$JAJ\in\rho(G)^{\prime}$. Let $g\in G$ be arbitrary. We claim that
$JAJ\rho_{g}=\rho_{g}JAJ$. Since $\\{\delta_{h}\\}_{h\in G}$ is a basis for
$\ell^{p}(G)$ it suffices to show
$JAJ\rho_{g}\delta_{h}=\rho_{g}JAJ\delta_{h}$, for all $h\in G$. Now noting
$A\in\lambda(G)^{\prime}$, we get

$\displaystyle JAJ\rho_{g}\delta_{h}$
$\displaystyle=JAJ\delta_{hg^{-1}}=JA\delta_{gh^{-1}}=JA\lambda_{g}\delta_{h^{-1}}=J\lambda_{g}A\delta_{h^{-1}}$
$\displaystyle=J\lambda_{g}\left(\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\delta_{u}\right)=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})J\lambda_{g}\delta_{u}$ $\displaystyle=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})J\delta_{gu}=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\delta_{u^{-1}g^{-1}}$

and

$\displaystyle\rho_{g}JAJ\delta_{h}$
$\displaystyle=\rho_{g}JA\delta_{h^{-1}}=\rho_{g}J\left(\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\delta_{u}\right)=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\rho_{g}J\delta_{u}$ $\displaystyle=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\rho_{g}\delta_{u^{-1}}=\sum_{u\in
G}\zeta_{u}(A\delta_{h^{-1}})\delta_{u^{-1}g^{-1}}.$

Clearly $\Phi$ is linear. Since $J^{2}=I_{\ell^{p}(G)}$, $\Phi$ is
multiplicative. Through a direct calculation we see that inverse of $\Phi$ is
the map $\lambda(G)^{\prime\prime}\ni B\mapsto JBJ\in\rho(G)^{\prime\prime}$.
∎

Following is the most important result for generators of group-frames for
Banach spaces.

######  Theorem 2.17.

Let $\pi:G\to\mathcal{II}(\mathcal{X})$ be a invertible isometric
representation of a group $G$ which admits a functional-vector $(f,\tau)$.
Then

(14) $\displaystyle
f[\mathbb{I}^{*}(\pi(G)^{\prime\prime})^{-1},\mathbb{I}^{*}(\pi(G)^{\prime\prime})]\tau\subseteq\mathscr{F}_{G}(\pi,\mathcal{X}).$

###### Proof.

Let $g\in G$ and define
$\Delta_{g}\coloneqq\lambda_{g}|_{{\theta_{f}(\mathcal{X})}}$. Theorem 2.8
says that $\pi$ is invertible isometrically equivalent to the representation
$\Delta$ with functional-vector
$(\zeta_{e}(\theta_{f}\theta_{\tau}),\theta_{f}\theta_{\tau}\delta_{e})$.
Therefore, without loss of generality we may assume that

$\displaystyle\mathcal{X}=\theta_{f}(\mathcal{X}),\quad\pi=\Delta,\quad
G=\\{\lambda_{g}|{{}_{\theta_{f}(\mathcal{X})}}:g\in
G\\},\quad\tau=\theta_{f}\theta_{\tau}\delta_{e},\quad
f=\zeta_{e}(\theta_{f}\theta_{\tau}).$

Let $U:\theta_{f}(\mathcal{X})\to\theta_{f}(\mathcal{X})$ be an invertible
isometry such that $U\in\mathbb{I}^{*}(\pi(G)^{\prime\prime})$. We need to
show that $(fU^{-1},U\tau)\in\mathscr{F}_{G}(\pi,\mathcal{X})$. Define
$\tilde{\tau}\coloneqq U\tau$, $\tilde{\tau_{g}}\coloneqq\pi_{g}\tilde{\tau}$
and $\tilde{f}\coloneqq fU^{-1}$,
$\tilde{f_{g}}\coloneqq\tilde{f}\pi_{g^{-1}}$ for all $g\in G$. To prove the
theorem, now it suffices to show that $(\\{\tilde{f_{g}}\\}_{g\in
G},\\{\tilde{\tau_{g}}\\}_{g\in G})$ is an group-p-USF for $\mathcal{X}$. Let
$x=\theta_{f}y\in\theta_{f}(\mathcal{X})$. Using Theorem 2.15 and Theorem
2.16, we may assume that
$U\in\\{\lambda_{g}|{{}_{\theta_{f}(\mathcal{X})}}:g\in G\\}^{\prime}$. Since
$\theta_{f}\theta_{\tau}z=z$ for all $z\in\theta_{f}(\mathcal{X})$ and
$U:\theta_{f}(\mathcal{X})\to\theta_{f}(\mathcal{X})$, we then have
$\theta_{f}\theta_{\tau}U=U$. Therefore

$\displaystyle\sum_{{g\in G}}\tilde{f_{g}}(x)\tilde{\tau_{g}}$
$\displaystyle=\sum_{{g\in
G}}(\tilde{f}\pi_{g^{-1}})(x)\pi_{g}\tilde{\tau}=\sum_{{g\in
G}}((fU^{-1})\pi_{g^{-1}})(x)\pi_{g}U\tau$ $\displaystyle=\sum_{{g\in
G}}f(U^{-1}\pi_{g^{-1}}x)\pi_{g}U\tau=\sum_{{g\in
G}}(\zeta_{e}\theta_{f}\theta_{\tau})(U^{-1}\pi_{g^{-1}}x)\pi_{g}U\theta_{f}\theta_{\tau}\delta_{e}$
$\displaystyle=\sum_{{g\in
G}}\zeta_{e}(\theta_{f}\theta_{\tau}U^{-1}\pi_{g^{-1}}x)\pi_{g}U\theta_{f}\theta_{\tau}\delta_{e}=\sum_{{g\in
G}}\zeta_{e}(\theta_{f}\theta_{\tau}U^{-1}\lambda_{g^{-1}}x)\lambda_{g}U\theta_{f}\theta_{\tau}\delta_{e}$
$\displaystyle=\sum_{{g\in
G}}\zeta_{e}(\theta_{f}\theta_{\tau}\lambda_{g^{-1}}U^{-1}x)U\lambda_{g}\theta_{f}\theta_{\tau}\delta_{e}=\sum_{{g\in
G}}\zeta_{e}(\lambda_{g^{-1}}\theta_{f}\theta_{\tau}U^{-1}x)U\theta_{f}\theta_{\tau}\lambda_{g}\delta_{e}$
$\displaystyle=\sum_{{g\in
G}}\zeta_{g}(\theta_{f}\theta_{\tau}U^{-1}x)U\theta_{f}\theta_{\tau}\delta_{g}=U\theta_{f}\theta_{\tau}\theta_{f}\theta_{\tau}U^{-1}x$
$\displaystyle=U\theta_{f}\theta_{\tau}U^{-1}x=UU^{-1}x=x.$

∎

Han and Larson characterized (with an equality) frame vectors for Hilbert
spaces (see Theorem 6.17 in [16]). Later Kaftal, Larson and Zhang noticed that
the set of all frame vectors is path-connected in norm topology (see Theorem
8.1 in [20]). Based on these, we raise the following questions.

######  Problem 2.18.

Given an invertible isometric representation
$\pi:G\to\mathbb{I}^{*}(\mathcal{X})$ which admits a functional-vector
$(f,\tau)$, characterize $\mathscr{F}_{G}(\pi,\mathcal{X})$. In particular,
classify Banach spaces, groups and representations such that

  1. (i)

$f[\mathbb{I}^{*}(\pi(G)^{\prime})^{-1},\mathbb{I}^{*}(\pi(G)^{\prime})]\tau=\mathscr{F}_{G}(\pi,\mathcal{X})$.

  2. (ii)

$f[\mathbb{I}^{*}(\pi(G)^{\prime\prime})^{-1},\mathbb{I}^{*}(\pi(G)^{\prime\prime})]\tau=\mathscr{F}_{G}(\pi,\mathcal{X})$.

######  Problem 2.19.

Is $\mathscr{F}_{G}(\pi,\mathcal{X})\subseteq\mathcal{X}^{*}\times\mathcal{X}$
path connected in the norm-topology?

##  3\. Finite Gabor-Schauder frames

In this section we study Schauder frame generated by time-frequency shifts on
finite abelian groups for finite dimensional Banach space $\mathbb{C}^{o(G)}$.
Our main motivation comes from the Gabor analysis on finite abelian groups
presented by Pfander in [27] and Feichtinger, Kozek and Luef in [11]. Let $G$
be a finite abelian group with identity $e$, $o(G)$ be the order of $G$ and
$\widehat{G}$ be the set of all characters of $G$. We denote the circle group
by $\mathbb{T}$. Then $\widehat{G}$ becomes a group with respect to pointwise
multiplication of characters. The character which sends every element of $G$
to 1 is called as identity character and is denoted by $1_{G}$. Let
$\\{\delta_{g}\\}_{g\in G}$ be the standard basis for $\mathbb{C}^{o(G)}$ and
$\\{\zeta_{g}\\}_{g\in G}$ be the coordinate functionals associated with
$\\{\delta_{g}\\}_{g\in G}$. For $x=(x_{g})_{g\in G},y=(y_{g})_{g\in
G}\in\mathbb{C}^{o(G)}$, we set

$\displaystyle x\cdot y\coloneqq(x_{g}y_{g})_{g\in G}$

and

$\displaystyle x^{*}\coloneqq(\overline{x_{g}})_{g\in G}.$

From the classical Fourier analysis on finite abelian groups, we note that we
have the following properties (see [24]).

  1. (1)

$o(G)=o(\widehat{G})$.

  2. (2)

If $\xi,\chi\in\widehat{G}$, then $\langle\xi,\chi\rangle=\delta_{\xi,\chi}$,
where

(15) $\displaystyle\langle\xi,\chi\rangle\coloneqq\frac{1}{o(G)}\sum_{g\in
G}\xi(g)\overline{\chi(g)}.$

  3. (3)

If $g,h\in G$, then $\langle g,h\rangle=\delta_{g,h}$, where

(16) $\displaystyle\langle
g,h\rangle\coloneqq\frac{1}{o(\widehat{G})}\sum_{\xi\in\widehat{G}}\xi(g)\overline{\xi(h)}.$

Given $\xi\in\widehat{G}$, the modulation operator
$M_{\xi}:\mathbb{C}^{o(G)}\to\mathbb{C}^{o(G)}$ is defined by

$\displaystyle M_{\xi}(x_{g})_{g\in G}\coloneqq(\xi(g)x_{g})_{g\in
G},\quad\forall(x_{g})_{g\in G}\in\mathbb{C}^{o(G)}.$

Given $k\in G$, the translation operator
$T_{k}:\mathbb{C}^{o(G)}\to\mathbb{C}^{o(G)}$ by

$\displaystyle T_{k}(x_{g})_{g\in G}\coloneqq(x_{g-k})_{g\in
G},\quad\forall(x_{g})_{g\in G}\in\mathbb{C}^{o(G)}.$

Let $\lambda\coloneqq(k,\xi)\in G\times\widehat{G}$. By composing modulation
operator and translation operator we get the time-frequency shift operator
$\pi(k,\xi):\mathbb{C}^{o(G)}\to\mathbb{C}^{o(G)}$ defined as

$\displaystyle\pi(\lambda)=\pi(k,\xi)\coloneqq M_{\xi}T_{k}.$

Thus the action of $\pi(\lambda)$ is given by

$\displaystyle\pi(\lambda)(x_{g})_{g\in G}=(\xi(g)x_{g-k})_{g\in
G},\quad\forall(x_{g})_{g\in G}\in\mathbb{C}^{o(G)}.$

Following properties of time-frequency shift operators will be used and are
well-known.

######  Theorem 3.1.

[11] Let $\lambda=(k,\xi),\mu=(l,\chi)\in G\times\widehat{G}$. Then

  1. (i)

$\pi(\lambda+\mu)=\chi(k)\pi(\lambda)\pi(\mu)$.

  2. (ii)

$\pi(\lambda)\pi(\mu)=\overline{\chi(k)}\xi(l)\pi(\mu)\pi(\lambda)$.

  3. (iii)

$\pi(\lambda)^{-1}=\overline{\xi(k)}\pi(-\lambda)$.

Motivated from discrete Gabor analysis over finite abelian groups (see [27,
11]) we set the following notion. We emphasis here that, even though it is
true that $\mathbb{C}^{o(G)}$ is a Hilbert space, given a Banach space
structure on it, thinking of $\mathbb{C}^{o(G)}$ as a Hilbert space in frame
theory will not work. A recent influential instance is in defining the notion
of ‘Frame Potential’ for Banach spaces where usual direct generalization of
Hilbert space frame potential failed (see Proposition 2.5 in [6]).

######  Definition 3.2.

Given a subgroup $\Lambda\subseteq G\times\widehat{G}$, a nonzero
$f\in(\mathbb{C}^{o(G)})^{*}$ and a nonzero $\tau\in\mathbb{C}^{o(G)}$, the
pair
$(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$
is called as a Gabor-Schauder system. If the operator

$\displaystyle S_{f,\tau,\Lambda}:\mathbb{C}^{o(G)}\ni
x\mapsto\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}x)\pi(\lambda)\tau\in\mathbb{C}^{o(G)}$

is invertible, then
$(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$
is called as a Gabor-Schauder frame.

We are interested in subgroups $\Lambda$ of $G\times\widehat{G}$, which will
give Gabor-Schauder frames. First we show that the full lattice
$G\times\widehat{G}$ will give a Gabor-Schauder frame. Given a nonzero
$f\in(\mathbb{C}^{o(G)})^{*}$, we define analysis operator

$\displaystyle W_{f}:\mathbb{C}^{o(G)}\ni x\mapsto
W_{f}x\coloneqq(f(\pi(\lambda)^{-1}x)_{\lambda\in
G\times\widehat{G}}\in\mathbb{C}^{o(G\times\widehat{G})}$

and given a nonzero $\tau\in\mathbb{C}^{o(G)}$, we define synthesis operator

$\displaystyle
V_{\tau}:\mathbb{C}^{o(G\times\widehat{G})}\ni(a_{\lambda})_{\lambda\in
G\times\widehat{G}}\mapsto\sum_{\lambda\in
G\times\widehat{G}}a_{\lambda}\pi(\lambda)\tau\in\mathbb{C}^{o(G)}.$

Our first result is that composition of previous two operators give scalar
times identity.

######  Theorem 3.3.

For $f\in(\mathbb{C}^{o(G)})^{*}$ and $\tau\in\mathbb{C}^{o(G)}$,

(17) $\displaystyle V_{\tau}W_{f}=o(G)f(\tau)I_{\mathbb{C}^{o(G)}}.$

###### Proof.

Let $x=(x_{g})_{g\in G}\in\mathbb{C}^{o(G)}$ and $\tau=(\tau_{g})_{g\in G}$.
Then

$\displaystyle V_{\tau}W_{f}x=V_{\tau}(f[\pi(\lambda)^{-1}x])_{\lambda\in
G\times\widehat{G}}=\sum_{\lambda\in
G\times\widehat{G}}f[\pi(\lambda)^{-1}x]\pi(\lambda)\tau$
$\displaystyle=\sum_{(k,\xi)\in
G\times\widehat{G}}f[\overline{\xi(k)}\pi(-k,-\xi)(x_{g})_{g\in
G}]\pi(k,\xi)(\tau_{g})_{g\in G}$ $\displaystyle=\sum_{(k,\xi)\in
G\times\widehat{G}}f[\overline{\xi(k)}((-\xi)(h)x_{h+k})_{h\in
G})](\xi(g)\tau_{g-k})_{g\in G}=\sum_{(k,\xi)\in
G\times\widehat{G}}f[(\overline{\xi(k+h)}x_{h+k})_{h\in
G}](\xi(g)\tau_{g-k})_{g\in G}$ $\displaystyle=\left(\sum_{(k,\xi)\in
G\times\widehat{G}}f[(\overline{\xi(k+h)}x_{h+k})_{h\in
G}]\xi(g)\tau_{g-k}\right)_{g\in G}=\left(\sum_{(k,\xi)\in
G\times\widehat{G}}f[(\overline{\xi(k+h)}\xi(g)x_{h+k})_{h\in
G}]\tau_{g-k}\right)_{g\in G}$ $\displaystyle=\left(\sum_{k\in
G}\sum_{\xi\in\widehat{G}}f[(\overline{\xi(k+h)}\xi(g)x_{h+k})_{h\in
G}]\tau_{g-k}\right)_{g\in G}=\left(\sum_{k\in
G}f\left[\sum_{\xi\in\widehat{G}}(\overline{\xi(k+h)}\xi(g)x_{h+k})_{h\in
G}\right]\tau_{g-k}\right)_{g\in G}$ $\displaystyle=\left(\sum_{k\in
G}f\left[\sum_{\xi\in\widehat{G}}\overline{\xi(k+h)}\xi(g)x_{h+k}\right]_{h\in
G}\tau_{g-k}\right)_{g\in G}=o(G)\left(\sum_{k\in
G}f\left[\delta_{k+h,g}x_{h+k}\right]_{h\in G}\tau_{g-k}\right)_{g\in G}$
$\displaystyle=o(G)\sum_{g\in G}\sum_{k\in
G}f\left[\delta_{k+h,g}x_{h+k}\right]_{h\in
G}\tau_{g-k}\delta_{g}=o(G)\sum_{g\in G}\sum_{k\in G}f\left[\sum_{h\in
G}\delta_{k+h,g}x_{h+k}\delta_{h}\right]\tau_{g-k}\delta_{g}$
$\displaystyle=o(G)\sum_{g\in G}\sum_{k\in G}\sum_{h\in
G}\delta_{k+h,g}x_{h+k}f[\delta_{h}]\tau_{g-k}\delta_{g}=o(G)\sum_{g\in
G}\sum_{k\in G}x_{g}f[\delta_{g-k}]\tau_{g-k}\delta_{g}$
$\displaystyle=o(G)\sum_{g\in G}x_{g}f\left[\sum_{k\in
G}\tau_{g-k}\delta_{g-k}\right]\delta_{g}=o(G)\sum_{g\in
G}x_{g}f[(\tau_{k})_{k\in G}]\delta_{g}=o(G)f(\tau)x.$

∎

We call Equation (17) as Schauder-Moyal formula for Banach space. It is easy
to see that for Hilbert spaces, whenever $f$ is determined by $\tau$,
Schauder-Moyal formula reduces to familiar Moyal formula. Schauder-Moyal
formula immediately gives the following corollary.

######  Corollary 3.4.

If $f\in(\mathbb{C}^{o(G)})^{*}$ and $\tau\in\mathbb{C}^{o(G)}$ are such that
$f(\tau)\neq 0$, then

(18) $\displaystyle x=\frac{1}{o(G)f(\tau)}\sum_{\lambda\in
G\times\widehat{G}}f(\pi(\lambda)^{-1}x)\pi(\lambda)\tau,\quad\forall
x\in\mathbb{C}^{o(G)}.$

In other words,

$\displaystyle(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in
G\times\widehat{G}},\\{\pi(\lambda)\tau\\}_{\lambda\in G\times\widehat{G}})$

is a Gabor-Schauder frame for $\mathbb{C}^{o(G)}$.

We call Equation (18) as the inversion formula for Banach space. It reduces to
inversion formula for the short-time Fourier transform for Hilbert spaces,
whenever $f$ is determined by $\tau$.
Recall that given linear operators
$T,S:\mathbb{C}^{o(G)}\to\mathbb{C}^{o(G)}$, if we define

$\displaystyle\langle T,S\rangle_{\text{HS}}\coloneqq\sum_{g\in G}\langle
T\delta_{g},S\delta_{g}\rangle,$

then the space $\mathcal{L}(\mathbb{C}^{o(G)})$ of all linear operators from
$\mathbb{C}^{o(G)}$ to itself is a Hilbert space w.r.t. inner product $\langle
T,S\rangle_{\text{HS}}$. We denote this Hilbert space by
$\text{HS}(\mathbb{C}^{o(G)})$. We need the following result in continuation.

######  Theorem 3.5.

[11] The family

(19)
$\displaystyle\left\\{\frac{1}{\sqrt{o(G)}}\pi(\lambda)\right\\}_{\lambda\in
G\times\widehat{G}}$

is an orthonormal basis for $\text{HS}(\mathbb{C}^{o(G)})$.

Recall that given a subgroup (also known as lattice) $\Lambda\subseteq
G\times\widehat{G}$, we define the adjoint subgroup of $\Lambda$, denoted by
$\Lambda^{0}$ as

$\displaystyle\Lambda^{0}\coloneqq\\{\mu\in
G\times\widehat{G}:\pi(\lambda)\pi(\mu)=\pi(\mu)\pi(\lambda),\forall\lambda\in\Lambda\\}.$

Theorem 3.1 says that $\Lambda^{0}$ is a subgroup of $G\times\widehat{G}$. Now
given a nonzero vector $\tau\in\mathbb{C}^{o(G)}$ and a nonzero functional
$f\in(\mathbb{C}^{o(G)})^{*}$, we define the Gabor-Schauder frame operator
$S_{f,\tau,\Lambda}:\mathbb{C}^{o(G)}\to\mathbb{C}^{o(G)}$ as

$\displaystyle
S_{f,\tau,\Lambda}x\coloneqq\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}x)\pi(\lambda)\tau,\quad\forall
x\in\mathbb{C}^{o(G)}.$

Following key result will be used repeatedly. It mainly uses group properties
of $\Lambda$.

######  Theorem 3.6.

Let $\Lambda$ be a subgroup of $G\times\widehat{G}$,
$\tau\in\mathbb{C}^{o(G)}$ and $f\in(\mathbb{C}^{o(G)})^{*}$. Then for each
$\mu\in\Lambda$, the time-frequency shifts $\pi(\mu)$ commute with the Gabor-
Schauder frame operator $S_{f,\tau,\Lambda}$.

###### Proof.

Since $\pi(\mu)$ is invertible, to show
$\pi(\mu)S_{f,\tau,\Lambda}=S_{f,\tau,\Lambda}\pi(\mu)$ it suffices to show
that $\pi(\mu)^{-1}S_{f,\tau,\Lambda}\pi(\mu)=S_{f,\tau,\Lambda}$. Let
$x\in\mathbb{C}^{o(G)}$ and $\mu=(l,\chi)$. Now using the fact that $G$ is a
group and using Theorem 3.1,

$\displaystyle\pi(l,\chi)^{-1}S_{f,\tau,\Lambda}\pi(l,\chi)x$
$\displaystyle=\pi(l,\chi)^{-1}\left(\sum_{(k,\xi)\in\Lambda}f[\pi(k,\xi)^{-1}\pi(l,\chi)x]\pi(k,\xi)\tau\right)$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[\pi(k,\xi)^{-1}\pi(l,\chi)x]\pi(l,\chi)^{-1}\pi(k,\xi)\tau$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[(\pi(l,\chi)^{-1}\pi(k,\xi))^{-1}x]\pi(l,\chi)^{-1}\pi(k,\xi)\tau$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[(\overline{\chi(l)}\pi(-l,-\chi)\pi(k,\xi))^{-1}x]\overline{\chi(l)}\pi(-l,-\chi)\pi(k,\xi)\tau$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[(\pi(-l,-\chi)\pi(k,\xi))^{-1}x]\pi(-l,-\chi)\pi(k,\xi)\tau$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[(\overline{\xi(-l)}\pi(-l+k,-\chi+\xi))^{-1}x]\overline{\xi(-l)}\pi(-l+k,-\chi+\xi)\tau$
$\displaystyle=\sum_{(k,\xi)\in\Lambda}f[\pi(-l+k,-\chi+\xi)^{-1}x]\pi(-l+k,-\chi+\xi)\tau=S_{f,\tau,\Lambda}x.$

∎

Recall that (see[12]) a pair
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ is said to be an
approximate Schauder frame (written as ASF) for $\mathbb{C}^{o(G)}$ if the
operator

$\displaystyle S_{f,\tau}:\mathbb{C}^{o(G)}\ni x\mapsto
S_{f,\tau}x\coloneqq\sum_{j=1}^{n}f_{j}(x)\tau_{j}\in\mathbb{C}^{o(G)}$

is invertible. Also we recall that an ASF
$(\\{g_{j}\\}_{j=1}^{n},\\{\omega_{j}\\}_{j=1}^{n})$ for $\mathbb{C}^{o(G)}$
is said to be a dual for $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$ if

$\displaystyle
x=\sum_{j=1}^{n}f_{j}(x)\omega_{j}=\sum_{j=1}^{n}g_{j}(x)\tau_{j},\quad\forall
x\in\mathbb{C}^{o(G)}.$

It is an easy observation that the ASF
$(\\{f_{j}S_{f,\tau}^{-1}\\}_{j=1}^{n},\\{S_{f,\tau}^{-1}\tau_{j}\\}_{j=1}^{n})$
is always a dual for $(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$. This
is called as canonical dual of
$(\\{f_{j}\\}_{j=1}^{n},\\{\tau_{j}\\}_{j=1}^{n})$. Following corollary says
that canonical dual of Gabor-Schauder frame is again a Gabor-Schauder frame.

######  Corollary 3.7.

Let $\Lambda$ be a subgroup of $G\times\widehat{G}$,
$\tau\in\mathbb{C}^{o(G)}$ and $f\in(\mathbb{C}^{o(G)})^{*}$. If
$(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$
is a Gabor-Schauder frame for $\mathbb{C}^{o(G)}$, then its canonical dual can
be written as
$(\\{\phi(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\omega\\}_{\lambda\in\Lambda})$
for some $\omega\in\mathbb{C}^{o(G)}$ and $\phi\in(\mathbb{C}^{o(G)})^{*}$. In
other words, there exist $\omega\in\mathbb{C}^{o(G)}$ and
$\phi\in(\mathbb{C}^{o(G)})^{*}$ such that
$(\\{\phi(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\omega\\}_{\lambda\in\Lambda})$
is a Gabor-Schauder frame for $\mathbb{C}^{o(G)}$ and

$\displaystyle
x=\sum_{\lambda\in\Lambda}\phi(\pi(\lambda)^{-1}x)\pi(\lambda)\tau=\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}x)\pi(\lambda)\omega,\quad\forall
x\in\mathbb{C}^{o(G)}.$

###### Proof.

Canonical dual for
$(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$
is given by
$(\\{f(\pi(\lambda)^{-1}S_{f,\tau,\Lambda}^{-1})\\}_{\lambda\in\Lambda},\\{S_{f,\tau,\Lambda}^{-1}\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$.
Define $\omega\coloneqq S_{f,\tau,\Lambda}^{-1}\tau$ and $\phi\coloneqq
fS_{f,\tau,\Lambda}^{-1}$. Then using Theorem 3.6,

$\displaystyle\\{S_{f,\tau,\Lambda}^{-1}\pi(\lambda)\tau\\}_{\lambda\in\Lambda}=\\{\pi(\lambda)S_{f,\tau,\Lambda}^{-1}\tau\\}_{\lambda\in\Lambda}=\\{\pi(\lambda)\omega\\}_{\lambda\in\Lambda},$
$\displaystyle\\{f(\pi(\lambda)^{-1}S_{f,\tau,\Lambda}^{-1})\\}_{\lambda\in\Lambda}=\\{f(S_{f,\tau,\Lambda}^{-1}\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda}=\\{\phi(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda}.$

∎

Next result is the key result of this section.

######  Theorem 3.8.

Let $\Lambda$ be a subgroup of $G\times\widehat{G}$ and
$\tau\in\mathbb{C}^{o(G)}$, $f\in(\mathbb{C}^{o(G)})^{*}$. Then

(20) $\displaystyle
S_{f,\tau,\Lambda}x=\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}x)\pi(\lambda)\tau=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}}f(\pi(\mu)^{-1}\tau)\pi(\mu)x=\frac{o(\Lambda)}{o(G)}S_{f,x,\Lambda^{0}}\tau,\quad\forall
x\in\mathbb{C}^{o(G)}.$

###### Proof.

Theorem 3.5 says that

$\displaystyle S_{f,\tau,\Lambda}$ $\displaystyle=\sum_{\mu\in
G\times\widehat{G}}\left\langle
S_{f,\tau,\Lambda},\frac{1}{\sqrt{o(G)}}\pi(\mu)\right\rangle_{\text{HS}}\frac{1}{\sqrt{o(G)}}\pi(\mu)$
$\displaystyle=\sum_{\mu\in G\times\widehat{G}}c_{\mu}\pi(\mu),\quad\text{
where }c_{\mu}\coloneqq\frac{1}{o(G)}\left\langle
S_{f,\tau,\Lambda},\pi(\mu)\right\rangle_{\text{HS}},\forall\mu\in
G\times\widehat{G}.$

Let $\mu\in G\times\widehat{G}$ and $\lambda\in\Lambda$. Theorem 3.1 then
gives a $d_{\mu,\lambda}\in\mathbb{T}$ such that
$\pi(\lambda)^{-1}\pi(\mu)\pi(\lambda)=d_{\mu,\lambda}\pi(\mu)$. Using Theorem
3.6,

$\displaystyle\sum_{\mu\in G\times\widehat{G}}c_{\mu}\pi(\mu)$
$\displaystyle=\pi(\lambda)^{-1}S_{f,\tau,\Lambda}\pi(\lambda)=\pi(\lambda)^{-1}\left(\sum_{\mu\in
G\times\widehat{G}}c_{\mu}\pi(\mu)\right)\pi(\lambda)$
$\displaystyle=\sum_{\mu\in
G\times\widehat{G}}c_{\mu}\pi(\lambda)^{-1}\pi(\mu)\pi(\lambda)=\sum_{\mu\in
G\times\widehat{G}}c_{\mu}d_{\mu,\lambda}\pi(\mu).$

Therefore

$\displaystyle\sum_{\mu\in
G\times\widehat{G}}c_{\mu}(1-d_{\mu,\lambda})\pi(\mu)=0,\quad\forall\lambda\in\Lambda.$

Since $\\{\pi(\lambda)\\}_{\lambda\in G\times\widehat{G}}$ is linearly
independent (Theorem 3.5), we then have $c_{\mu}(1-d_{\mu,\lambda})=0$, for
all $\lambda\in\Lambda,$ for all $\mu\in G\times\widehat{G}$. Let
$\mu\notin\Lambda^{0}$. We claim that $c_{\mu}=0$. If this is not true, then
$1-d_{\mu,\lambda}=0$ for all $\lambda\in\Lambda$. But then we have
$\pi(\mu)\pi(\lambda)=\pi(\lambda)\pi(\mu)$ for all $\lambda\in\Lambda$ which
says $\mu\in\Lambda^{0}$ which is impossible. So claim holds. Hence the
formula for the Gabor-Schauder frame operator reduces to

(21) $\displaystyle
S_{f,\tau,\Lambda}=\sum_{\mu\in\Lambda^{0}}c_{\mu}\pi(\mu).$

Now let $\mu\in\Lambda^{0}$. Then using Equation (19),

$\displaystyle c_{\mu}$ $\displaystyle=\frac{1}{o(G)}\left\langle
S_{f,\tau,\Lambda},\pi(\mu)\right\rangle_{\text{HS}}=\frac{1}{o(G)}\sum_{g\in
G}\langle S_{f,\tau,\Lambda}\delta_{g},\pi(\mu)\delta_{g}\rangle$
$\displaystyle=\frac{1}{o(G)}\sum_{g\in
G}\left\langle\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}\delta_{g})\pi(\lambda)\tau,\pi(\mu)\delta_{g}\right\rangle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}\sum_{g\in
G}f(\pi(\lambda)^{-1}\delta_{g})\langle\pi(\lambda)\tau,\pi(\mu)\delta_{g}\rangle$
$\displaystyle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}\sum_{g\in
G}f(\pi(\lambda)^{-1}\delta_{g})\langle\pi(\mu)^{-1}\pi(\lambda)\tau,\delta_{g}\rangle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}\sum_{g\in
G}f\left[\langle\pi(\mu)^{-1}\pi(\lambda)\tau,\delta_{g}\rangle\pi(\lambda)^{-1}\delta_{g}\right]$
$\displaystyle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}f\left[\sum_{g\in
G}\langle\pi(\mu)^{-1}\pi(\lambda)\tau,\delta_{g}\rangle\pi(\lambda)^{-1}\delta_{g}\right]=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}f\left[\pi(\lambda)^{-1}\sum_{g\in
G}\langle\pi(\mu)^{-1}\pi(\lambda)\tau,\delta_{g}\rangle\delta_{g}\right]$
$\displaystyle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}\pi(\mu)^{-1}\pi(\lambda)\tau)=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}f(\pi(\mu)^{-1}\pi(\lambda)^{-1}\pi(\lambda)\tau)$
$\displaystyle=\frac{1}{o(G)}\sum_{\lambda\in\Lambda}f(\pi(\mu)^{-1}\tau)=\frac{o(\Lambda)}{o(G)}f(\pi(\mu)^{-1}\tau).$

By substituting the value of $c_{\mu}$ in Equation (21) finally gives

$\displaystyle
S_{f,\tau,\Lambda}=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}}f(\pi(\mu)^{-1}\tau)\pi(\mu).$

∎

We call Equation (20) as Fundamental Identity of Gabor-Schauder Frames (FIGSF)
or Schauder-Wexler-Raz identity or Schauder-Janssen representation of Gabor-
Schauder frame operator $S_{f,\tau,\Lambda}$. It represents the frame operator
corresponding to $\Lambda$ in terms of frame operator corresponding to
$\Lambda^{0}$. For Hilbert spaces it reduces to the usual Fundamental Identity
of Gabor Analysis (famously known as FIGA) derived in [27] and [11]. First
major consequence of FIGSF is the following criterion.

######  Theorem 3.9.

Let $\Lambda$ be a subgroup of $G\times\widehat{G}$ and
$\tau\in\mathbb{C}^{o(G)}$, $f\in(\mathbb{C}^{o(G)})^{*}$. Then

(22) $\displaystyle
x=\sum_{\lambda\in\Lambda}f(\pi(\lambda)^{-1}x)\pi(\lambda)\tau,\quad\forall
x\in\mathbb{C}^{o(G)}\iff
f(\pi(\mu)^{-1}\tau)=\frac{o(G)}{o(\Lambda)}\delta_{\mu,0},\quad\forall\mu\in\Lambda^{0}.$

###### Proof.

$(\Rightarrow)$ We have that the Gabor-Schauder frame operator
$S_{f,\tau,\Lambda}$ is identity operator. Theorem 3.8 then says that

$\displaystyle
I_{\mathbb{C}^{o(G)}}=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}}f(\pi(\mu)^{-1}\tau)\pi(\mu)=\frac{o(\Lambda)}{o(G)}f(\tau)I_{\mathbb{C}^{o(G)}}+\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}\setminus\\{(0,0)\\}}f(\pi(\mu)^{-1}\tau)\pi(\mu).$

By Theorem 3.5, the collection $\\{\pi(\lambda)\\}_{\lambda\in
G\times\widehat{G}}$ is linearly independent. Thus the validity of previous
equation gives

$\displaystyle f(\tau)=\frac{o(G)}{o(\Lambda)},\quad
f(\pi(\mu)^{-1}\tau)=0,\forall\mu\in\Lambda^{0}\setminus\\{(0,0)\\}.$

$(\Leftarrow)$ Using Equation (20),

$\displaystyle
S_{f,\tau,\Lambda}x=\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}}f(\pi(\mu)^{-1}\tau)\pi(\mu)x=\frac{o(\Lambda)}{o(G)}f(\tau)x+\frac{o(\Lambda)}{o(G)}\sum_{\mu\in\Lambda^{0}\setminus\\{(0,0)\\}}f(\pi(\mu)^{-1}\tau)\pi(\mu)x=x,\quad\forall
x\in\mathbb{C}^{o(G)}.$

∎

We call Equation (22) as Schauder-Wexler-Raz criterion. This is a
generalization of Wexler-Raz criterion derived in [27] and [11]. The criterion
says when we can say that certain pairs give Gabor-Schauder frames by checking
an algebraic equation on the adjoint lattice. We conclude by deriving the
following result, which we call partial Schauder Ron-Shen duality. Note that
for Hilbert spaces, the theorem is if and only if’ (see [27] and [11]).

######  Theorem 3.10.

Let $\Lambda$ be a subgroup of $G\times\widehat{G}$ and
$\tau\in\mathbb{C}^{o(G)}$, $f\in(\mathbb{C}^{o(G)})^{*}$. If
$(\\{f(\pi(\lambda)^{-1})\\}_{\lambda\in\Lambda},\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda})$
is a Gabor-Schauder frame for $\mathbb{C}^{o(G)}$, then both sets
$\\{\pi(\mu)^{-1}\tau\\}_{\mu\in\Lambda^{0}}$ and
$\\{f(\pi(\mu)^{-1})\\}_{\mu\in\Lambda^{0}}$ are linearly independent.

###### Proof.

Without loss of generality we may assume that
$S_{f,\tau,\Lambda}=I_{\mathbb{C}^{o(G)}}.$ Now Theorem 3.9 says that

$\displaystyle
f(\pi(\mu)^{-1}\tau)=\frac{o(G)}{o(\Lambda)}\delta_{\mu,0},\quad\forall\mu\in\Lambda^{0}.$

Let $\lambda,\mu\in\Lambda^{0}$ be such that $\lambda\neq\mu$. Then, since
$\Lambda^{0}$ is a group, we get that
$f(\pi(\lambda)^{-1}\pi(\mu)^{-1}\tau)=0$. We now suppose that
$\sum_{\mu\in\Lambda^{0}}c_{\mu}\pi(\mu)^{-1}\tau=0$ for some
$c_{\mu}\in\mathbb{C}$. Then for each $\lambda\in\Lambda^{0}$, we have

$\displaystyle
0=f\left(\pi(\lambda)^{-1}\left(\sum_{\mu\in\Lambda^{0}}c_{\mu}\pi(\mu)^{-1}\tau\right)\right)=\sum_{\mu\in\Lambda^{0}}c_{\mu}f(\pi(\lambda)^{-1}\pi(\mu)^{-1}\tau)=c_{\lambda}\frac{o(G)}{o(\Lambda)}.$

Therefore $\\{\pi(\lambda)\tau\\}_{\lambda\in\Lambda^{0}}$ is linearly
independent. On the other hand, let
$\sum_{\mu\in\Lambda^{0}}d_{\mu}f(\pi(\mu)^{-1})=0$ for some
$d_{\mu}\in\mathbb{C}$. Then for each $\lambda\in\Lambda^{0}$, we have

$\displaystyle
0=\sum_{\mu\in\Lambda^{0}}d_{\mu}f(\pi(\mu)^{-1}\pi(\lambda)^{-1}\tau)=d_{\lambda}\frac{o(G)}{o(\Lambda)}.$

Therefore $\\{f(\pi(\mu)^{-1})\\}_{\mu\in\Lambda^{0}}$ is linearly
independent. ∎

##  4\. Acknowledgments

Author thanks Prof. B.V. Rajarama Bhat for several useful discussions and
suggestions which improved the paper. Author is supported by J.C. Bose
Fellowship of Prof. B.V. Rajarama Bhat.

## References

  * [1] Akram Aldroubi, David Larson, Wai-Shing Tang, and Eric Weber.  Geometric aspects of frame representations of abelian groups.  Trans. Amer. Math. Soc., 356(12):4767–4786, 2004.
  * [2] Davide Barbieri, Eugenio Hernandez, and Javier Parcet.  Riesz and frame systems generated by unitary actions of discrete groups.  Appl. Comput. Harmon. Anal., 39(3):369–399, 2015.
  * [3] Marcin Bownik and Joseph W. Iverson.  Multiplication-invariant operators and the classification of LCA group frames.  J. Funct. Anal., 280(2):Paper No. 108780, 59, 2021.
  * [4] Peter G. Casazza, Deguang Han, and David R. Larson.  Frames for Banach spaces.  In The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), volume 247 of Contemp. Math., pages 149–182. Amer. Math. Soc., Providence, RI, 1999.
  * [5] Peter G. Casazza and Gitta Kutyniok, editors.  Finite frames: Theory and applications.  Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2013.
  * [6] J. A. Chávez-Domínguez, D. Freeman, and K. Kornelson.  Frame potential for finite-dimensional Banach spaces.  Linear Algebra Appl., 578:1–26, 2019.
  * [7] Ole Christensen and Say Song Goh.  Explicit construction of frames and pairs of dual frames on locally compact abelian groups.  Sampl. Theory Signal Process. Data Anal., 19(1):Paper No. 4, 26, 2021.
  * [8] R. J. Duffin and A. C. Schaeffer.  A class of nonharmonic Fourier series.  Trans. Amer. Math. Soc., 72:341–366, 1952.
  * [9] Dorin Ervin Dutkay, Deguang Han, and Gabriel Picioroaga.  Parseval frames for ICC groups.  J. Funct. Anal., 256(9):3071–3090, 2009.
  * [10] Yonina C. Eldar and Helmut Bölcskei.  Geometrically uniform frames.  IEEE Trans. Inform. Theory, 49(4):993–1006, 2003.
  * [11] Hans G. Feichtinger, Werner Kozek, and Franz Luef.  Gabor analysis over finite abelian groups.  Appl. Comput. Harmon. Anal., 26(2):230–248, 2009.
  * [12] D. Freeman, E. Odell, Th. Schlumprecht, and A. Zsák.  Unconditional structures of translates for $L_{p}(\mathbb{R}^{d})$.  Israel J. Math., 203(1):189–209, 2014.
  * [13] Karlheinz Gröchenig.  Describing functions: atomic decompositions versus frames.  Monatsh. Math., 112(1):1–42, 1991.
  * [14] Deguang Han.  Classification of finite group-frames and super-frames.  Canad. Math. Bull., 50(1):85–96, 2007.
  * [15] Deguang Han, Keri Kornelson, David Larson, and Eric Weber.  Frames for undergraduates, volume 40 of Student Mathematical Library.  American Mathematical Society, Providence, RI, 2007.
  * [16] Deguang Han and David R. Larson.  Frames, bases and group representations.  Mem. Amer. Math. Soc., 147(697):x+94, 2000.
  * [17] James R. Holub.  Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces.  Proc. Amer. Math. Soc., 122(3):779–785, 1994.
  * [18] Joseph W. Iverson.  Frames generated by compact group actions.  Trans. Amer. Math. Soc., 370(1):509–551, 2018.
  * [19] Kenneth W. Johnson.  Group matrices, group determinants and representation theory, volume 2233 of Lecture Notes in Mathematics.  Springer, Cham, 2019.
  * [20] Victor Kaftal, David R. Larson, and Shuang Zhang.  Operator-valued frames.  Trans. Amer. Math. Soc., 361(12):6349–6385, 2009.
  * [21] K. Mahesh Krishna and P. Sam Johnson.  Towards characterizations of approximate Schauder frame and its duals for Banach spaces.  J. Pseudo-Differ. Oper. Appl., 12(1):Paper No. 9, 13, 2021.
  * [22] Zhongyan Li and Deguang Han.  Frame vector multipliers for finite group representations.  Linear Algebra Appl., 519:191–207, 2017.
  * [23] Robert P. Mendez, Bernhard G. Bodmann, Zachery J. Baker, Micah G. Bullock, and Jacob E. McLaney.  Binary Parseval frames from group orbits.  Linear Algebra Appl., 556:265–300, 2018.
  * [24] Melvyn B. Nathanson.  Elementary methods in number theory, volume 195 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2000.
  * [25] Vignon Oussa.  Frames arising from irreducible solvable actions I.  J. Funct. Anal., 274(4):1202–1254, 2018.
  * [26] Vignon Oussa.  Compactly supported bounded frames on Lie groups.  J. Funct. Anal., 277(6):1718–1762, 2019.
  * [27] Götz E. Pfander.  Gabor frames in finite dimensions.  In Finite frames, Appl. Numer. Harmon. Anal., pages 193–239. Birkhäuser/Springer, New York, 2013.
  * [28] Kjetil Roysland.  Frames generated by actions of countable discrete groups.  Trans. Amer. Math. Soc., 363(1):95–108, 2011.
  * [29] Wai Shing Tang and Eric Weber.  Frame vectors for representations of abelian groups.  Appl. Comput. Harmon. Anal., 20(2):283–297, 2006.
  * [30] Richard Vale and Shayne Waldron.  Tight frames generated by finite nonabelian groups.  Numer. Algorithms, 48(1-3):11–27, 2008.
  * [31] Richard Vale and Shayne Waldron.  The symmetry group of a finite frame.  Linear Algebra Appl., 433(1):248–262, 2010.
  * [32] Richard Vale and Shayne Waldron.  The construction of $G$-invariant finite tight frames.  J. Fourier Anal. Appl., 22(5):1097–1120, 2016.
  * [33] Shayne Waldron and Nick Hay.  On computing all harmonic frames of $n$ vectors in $\mathbb{C}^{d}$.  Appl. Comput. Harmon. Anal., 21(2):168–181, 2006.
  * [34] Shayne F. D. Waldron.  An introduction to finite tight frames.  Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2018.