The density of Gabor systems in expansible locally compact abelian groups

TL;DR

This paper explores the properties of Gabor systems in expansible locally compact abelian groups, establishing density conditions for reproducing properties and linking modulation and Wiener spaces.

Contribution

It extends the concept of Beurling density to expansible groups and proves new embedding and Bessel conditions for Gabor systems in this context.

Findings

Density conditions characterize Gabor system properties.

Modulation spaces embed into Wiener spaces for certain groups.

A new proof of density results for Gabor frames is provided.

Abstract

We investigate the reproducing properties of Gabor systems within the context of expansible groups. These properties are established in terms of density conditions. The concept of density that we employ mirrors the well-known Beurling density defined in Euclidean space, which is made possible due to the expansive structure. Along the way, for groups with a compact open subgroup, we demonstrate that modulation spaces are continuously embedded in Wiener spaces. Utilizing this result, we derive the Bessel condition of Gabor systems. We also provide a straightforward proof of the density result for Gabor frames, utilizing a comparison theorem for coherent frames.