More about continuous Gabor frames on locally compact abelian groups

TL;DR

This paper investigates the conditions under which continuous Gabor frames can be generated on second countable locally compact abelian groups, extending existing transforms and providing new characterizations using fiberization techniques.

Contribution

It reformulates the Zak transform for integer-oversampled lattices under broader subgroup assumptions and characterizes continuous Gabor frames via fiberization and frame families in $l^{2}(\widehat{H^{ot}})$.

Findings

Necessary and sufficient conditions for continuous Gabor frames on LCA groups.

Generalization of the Zak transform for closed subgroup translation and modulation groups.

Characterization of frames using fiberization and family of frames in $l^{2}(\widehat{H^{ot}})$.

Abstract

For a second countable locally compact abelian (LCA) group $G$, we study some necessary and sufficient conditions to generate continuous Gabor frames for $L^{2}(G)$. To this end, we reformulate the generalized Zak transform proposed by Grochenig in the case of integer-oversampled lattices, however our formulation rely on the assumption that both translation and modulation groups are only closed subgroups. Moreover, we discuss the possibility of such generalization and apply several examples to demonestrate the necessity of standing conditions in the results. Finally, by using the generalized Zak transform and fiberization technique, we obtain some characterization of continuous Gabor frames for $L^{2}(G)$ in term of a family of frames in $l^{2}(\widehat{H^{\perp}})$ for a closed co-compact subgroup $H$ of $G$.