Co-compact Gabor Systems on Locally Compact Abelian Groups

Abstract

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

Duality results for co-compact Gabor systems

“…To be more precise, a co-compact Gabor system is a Bessel family if, and only if, the adjoint Gabor system is a Bessel sequence. The result generalizes a result in [14]; a proof can be found in [10].…”

“…The duality results go back to a series of papers in the 1990s [1]- [9] on discrete Gabor systems in L 2 (R) and L 2 (R d ) with modulations and translations along lattices. Recently, these duality results were extended by the two authors [10] to the setting of Gabor systems in L 2 (G) with modulations and translations along closed, co-compact subgroups, where G is a second countable locally compact abelian (LCA) group. The aim of this note is to give an overview of these extended duality results.…”

Density and duality theorems for regular Gabor frames

“…The duality principle can now be proven using general frame theory; the proof strategy is similar to the proof of the duality principle for separable, co-compact subgroups in [28]. Let us comment on a difference between the Bessel duality and the duality principle.…”

“…The Wexler-Raz biorthogonality relations were previously available for non-separable, uniform lattices ∆ ⊂ G × G on elementary LCA groups G = R n × T ℓ × Z k × F m by the work of Feichtinger and Kozek [14], while the duality principle (formulated without bounds) was proven by Feichtinger and Zimmermann [16] for Gabor systems G (g, ∆) in L 2 (R n ) with ∆ being a non-separable, full-rank lattice in R 2n . The authors proved in [28] both the Wexler-Raz biorthogonality relations and the duality principle on LCA groups for separable, co-compact subgroups ∆ = Λ × Γ ⊂ G × G using the theory of translation invariant systems; an approach that does not generalize to the non-separable case.…”

On Partition of Unities Generated by Entire Functions and Gabor Frames in $$ L^2({\mathbb R}^d) $$ L 2 ( R d ) and $$\ell ^2({\mathbb Z}^d)$$ ℓ 2 ( Z d )

“…For further information on discrete Gabor systems we refer to the paper [12] by Janssen (which also deals with the more general case of shift-invariant systems), [4,5] by Cvetković and Vetterli, as well as to the recent paper [19] by Lopez and Han. We also mention that the theory for translation invariant systems on LCA groups yields a joint framework to Gabor theory on L 2 (R d ) and 2 (Z d ), see, e.g., the papers [13,14] by Jakobsen and Lemvig.…”