The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

“…Assume that H is a discrete subgroup. It follows that µ G (X) is finite if, and only if, H is cocompact, i.e., H is a uniform lattice [7]. From [7], we also have that the mapping x → x + H from (X, µ G ) to (G/H, µ G/H ) is measure-preserving, and the mapping Q(f ) = f ′ defined by…”

“…Before we focus on Gabor systems, let us first show some results concerning the class of translation invariant systems, recently introduced in [7,29], which contains the class of (semi) co-compact Gabor systems. We define translation invariant systems as follows.…”

“…However, we begin our work on Gabor systems with a study of semi co-compact Gabor systems as special cases of co-compact translation invariant systems, recently introduced in [7,29]. For translation invariant systems we consider fiberization characterization of frames for translation invariant subspaces (Theorem 3.1), generalizing results from [5,7,8,39].…”

“…For translation invariant systems we consider fiberization characterization of frames for translation invariant subspaces (Theorem 3.1), generalizing results from [5,7,8,39]. Using these fiberization techniques we will develop Zak transform methods for Gabor analysis in L 2 (G).…”

“…For related work on locally compact (abelian) groups we refer to the recent papers [2,3,7,8,13,18,29,34] as well as the book [20] and the references therein.…”