Duality of Gabor frames and Heisenberg modules

TL;DR

This paper explores the duality between Gabor frames and Heisenberg modules, showing how time-frequency analysis tools reveal the structure of these modules and their relation to Gabor systems, including existence results for Gabor frames.

Contribution

It establishes a novel connection between Heisenberg modules and Gabor frames using the Feichtinger algebra, demonstrating duality principles and conditions for Gabor frame generation.

Findings

Feichtinger algebra acts as an equivalence bimodule between certain $C^*$-algebras.

Heisenberg modules are finitely generated and projective for co-compact subgroups.

Existence of Gabor frames generated by a single atom for non-rational lattices with volume less than one.

Abstract

Given a locally compact abelian group $G$ and a closed subgroup $\Lambda$ in $G\times\widehat{G}$, Rieffel associated to $\Lambda$ a Hilbert $C^*$-module $\mathcal{E}$, known as a Heisenberg module. He proved that $\mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(\Lambda,\textsf{c})$ and $C^*(\Lambda^\circ,\bar{\textsf{c}})$, where $\Lambda^{\circ}$ denotes the adjoint subgroup of $\Lambda$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra ${\textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${\textbf{S}}_{0}(\Lambda,\textsf{c})$ and ${\textbf{S}}_{0}(\Lambda^{\circ},\bar{\textsf{c}})$ of $C^*(\Lambda,\textsf{c})$ and $C^*(\Lambda^\circ,\bar{\textsf{c}})$, respectively. Further, we prove that ${\textbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $\Lambda$. In this case the generators $g_1,\ldots,g_n$ of the left ${\textbf{S}}_{0}(\Lambda)$-module ${\textbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,\ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $\Lambda^{\circ}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $\Lambda$ in $\mathbb{R}^{2m}$ with volume ${s}(\Lambda)<1$ there exists a Gabor frame generated by a single atom in ${\textbf{S}}_{0}(\mathbb{R}^m)$.