Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

Abstract

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let Γ j , j ∈ J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group G and study systems of the form ∪ j∈J {g j,p (· − γ)} γ∈Γj,p∈Pj with generators g j,p in L 2 (G) and with each P j being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical α local integrability condition (α-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for L 2 (G). This generalizes results on generalized shift invariant systems, where each P j is assumed to be countable and each Γ j is a uniform lattice in G, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices Γ j , our characterizations improve known results since the class of GTI systems satisfying the α-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for L 2 (G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in L 2 (R n ) are special cases of the same general characterizing equations.

Co-compact Gabor Systems on Locally Compact Abelian Groups

“…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.…”

“…The nature of these assumptions are discussed in [29]. Observe that any closed subgroup P of G (or G) with the Haar measure is admissible if p → g p is continuous, e.g.,…”

“…By a result on so-called characterizing equations from [29], we now characterize when two semi co-compact Gabor systems are dual frames. Using the equivalence of frame properties for systems {E γ T λ g} λ∈Λ,γ∈Γ and {T γ F −1 T λ g} γ∈Γ,λ∈Λ with generator g ∈ L 2 (G) yields the following characterizing equations in the time domain.…”

“…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 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.…”