Co-compact Gabor systems on locally compact abelian groups

TL;DR

This paper extends Gabor analysis to locally compact abelian groups, providing new duality, characterization, and non-existence results for Gabor frames without relying on discreteness or co-compactness assumptions.

Contribution

It introduces a unified framework for continuous and discrete Gabor frames on LCA groups, including new duality principles and characterization results.

Findings

Characterization of rational oversampling in LCA groups

Non-existence of critically sampled continuous Gabor frames

Walnut and Janssen representations for Gabor frame operators

Abstract

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.