Density and duality theorems for regular Gabor frames

TL;DR

This paper extends density and duality theorems for Gabor frames to non-separable, closed subgroups of the phase space, broadening classical results and providing new insights even in Euclidean settings.

Contribution

It generalizes classical density theorems and duality principles for Gabor frames to include non-separable, closed subgroups of the phase space.

Findings

Density theorems for general closed subgroups are established.

Classical Wexler-Raz biorthogonal relations are extended.

Duality principles are generalized to non-separable subgroups.

Abstract

We investigate Gabor frames on locally compact abelian groups with time-frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the "size" of the subgroup. From these density results we are able to extend the classical Wexler-Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time-frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.