Sampling and periodization of generators of Heisenberg modules

Abstract

This paper considers generators of Heisenberg modules in the case of twisted group C * -algebras of closed subgroups of locally compact abelian groups and how the restrction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of locally compact abelian groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov-Hausdorff distance to the irrational noncommutative torus.

A Brief Introduction to the Feichtinger Algebra $\mathbf{S}_{0}(\mathbb{R})$

“…We refer the reader to [11] for interesting historical remarks and references to various applications of the Feichtinger algebra. For more recent applications we recommend [2,3,4,18,22,25].…”

Deformations and Balian–Low theorems for Gabor frames on the adeles

“…Gabor analysis on other groups appear naturally in applications. For instance, the integers Z, finite cyclic groups Z/nZ and the circle group T are important when sampling or periodizing Gabor frames on R [29,32,33,42]. Other considerations on general locally compact abelian groups include [25,29,35], and Balian-Low type phenomena were proved to hold for finite groups in [36].…”

Deformations of Gabor frames on the adeles and other locally compact abelian groups

“…Gabor analysis on other groups appear naturally in applications. For instance, the integers Z, finite cyclic groups Z/nZ and the circle group T are important when sampling or periodizing Gabor frames on R [26,29,30,39]. Other considerations on general locally compact abelian groups include [21,26,32], and Balian-Low type phenomena were proved to hold for finite groups in [33].…”