Identifying continuous Gabor frames on locally compact Abelian groups

Abstract

In this paper, we establish some necessary and sufficient conditions for constructing continuous Gabor frames in L2(G), where G is a second countable locally compact abelian (LCA) group. More precisely, we reformulate the generalized Zak transform defined by A. Weil on LCA groups and later proposed by Gr?chenig in the case of integer-oversampled lattices, however our approach is regarding the assumption that both translation and modulation groups are closed subgroups. Moreover, we discuss the possibility of such a generalization and apply several examples to demonstrate the necessity of standing conditions in the results. Finally, by using the generalized Zak transform and fiberization technique, we characterize the continuous Gabor frames of L2(G) in terms of a family of frames in l2(cH?) for a closed co-compact subgroup H of G.