Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups

TL;DR

This paper introduces reproducing pairs as a generalization of continuous frames, investigates their properties on LCA groups, and applies the theory to systems generated by affine Weyl-Heisenberg group actions, highlighting structural insights and limitations.

Contribution

It generalizes continuous frames to reproducing pairs on LCA groups and analyzes their structure, including the frame operator, with applications to affine Weyl-Heisenberg systems.

Findings

Reproducing pairs allow bounded analysis and synthesis without the frame condition.

Sufficient conditions for systems to form continuous frames or reproducing pairs are provided.

Existence of systems with continuous frames but no dual with the same structure is demonstrated.

Abstract

In this paper we introduce and investigate the concept of repro- ducing pairs as a generalization of continuous frames. Reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages. Moreover, we will investigate certain continuous frames (resp. reproducing pairs) on LCA groups, which can be described as a continuous version of nonstationary Ga- bor systems and state sufficient conditions for these systems to form a continuous frame (resp. reproducing pair). As a byproduct we iden- tify the structure of the frame operator (resp. resolution operator). We will apply our results to systems generated by a unitary action of a subset of the affine Weyl-Heisenberg group in $L^2(\mathbb{R})$. This setup will also serve as a nontrivial example of a system for which, whereas continuous frames exist, no dual system with the same structure exists even if we drop the frame property.