IDENTIFYING CONTINUOUS GABOR FRAMES ON LOCALLY COMPACT ABELIAN GROUPS

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 approach is regarding 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 demonstrate the necessity the standing conditions in the results‎. ‎Finally‎, ‎by using the generalized Zak transform and fiberization technique‎, ‎we obtain some characterizations of continuous Gabor frames for L^{2}(G) in terms of a family of frames in l^{2}(\widehat{H^{\perp}}) for a closed co-compact subgroup H of G‎.