Duality of Gabor frames and Heisenberg modules

Abstract

Given a locally compact abelian group $G$ and a closed subgroup $\Lambda$ in $G\times \hat{G}$, Rieffel associated to $\Lambda$ a Hilbert $C^{\ast }$-module $\mathcal{E}$, known as a Heisenberg module. He proved that $\mathcal{E}$ is an equivalence bimodule between the twisted group $C^{\ast }$-algebra $C^{\ast }(\Lambda ,\mathsf{c})$ and $C^{\ast }({\Lambda }^{\circ },\bar{\mathsf{c}})$, 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 ${\mathbb{S}}_O(G)$ is an equivalence bimodule between the Banach subalgebras ${\mathbb{S}}_O(\Lambda ,\mathsf{c})$ and ${\mathbb{S}}_O({\Lambda }^{\circ },\bar{\mathsf{c}})$ of $C^{\ast }(\Lambda ,\mathsf{c})$ and $C^{\ast }({\Lambda }^{\circ },\bar{\mathsf{c}})$, respectively. Further, we prove that ${\mathbb{S}}_O(G)$ is finitely generated and projective exactly for co-compact closed subgroups $\Lambda$. In this case the generators $g_1,\dots ,g_n$ of the left ${\mathbb{S}}_O(\Lambda )$-module ${\mathbb{S}}_O(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,\dots ,g_n$ 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 ${\Lambda }^{\circ }({\mathbb{R}}^m)$.