The continuous nonstationary Gabor transform on LCA groups with applications to representations of the affine Weyl-Heisenberg group

TL;DR

This paper introduces a unifying framework for continuous nonstationary Gabor systems on LCA groups, explores conditions for these systems to form continuous frames or reproducing pairs, and examines dual systems with identity resolution operators, providing a counterexample involving the affine Weyl-Heisenberg group.

Contribution

It generalizes continuous frames via reproducing pairs on LCA groups and investigates their structure and duality properties, including a counterexample related to the affine Weyl-Heisenberg group.

Findings

Characterization of continuous nonstationary Gabor systems as continuous frames or reproducing pairs.

Identification of the structure of the frame and resolution operators.

Counterexample showing not all systems admit duals with identity resolution operator.

Abstract

In this paper we introduce and investigate the concept of reproducing pairs which generalizes continuous frames. We will introduce a concept that represents a unifying way to look at certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as continuous nonstationary Gabor systems and investigate conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we identify the structure of the frame operator (resp. resolution operator). Moreover, we ask the question, whether there always exist mutually dual systems with the same structure such that the resolution operator is given by the identity, i.e. given $A:X\rightarrow B(\mathcal{H})$, if there exist $\psi,\varphi\in\mathcal{H}$, s.t. \begin{equation*} f=\int_X \langle f,A(x)\psi\rangle A(x)\varphi d\mu(x),\ \ \forall f\in \mathcal{H} \end{equation*} and show that the answer is not affirmative. As a counterexample we use a system generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$.