The Continuous Zak Transform and Generalized Gabor Frames

Let G be a locally compact abelian group and H be a closed (not necessarily discrete) subgroup of G. In this article, we introduce the notion of Zak transform associated to H and obtain a necessary and sufficient condition to generate continuous Gabor frames for L ²(G). These results can be extended to non-abelian locally compact groups which are semidirect products. As an application, we obtain a characterization of admissible vectors for the regular and quasi regular representations.