More about 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 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 $G$.