The Balian-Low theorem for locally compact abelian groups and vector bundles

Abstract

Let $\Lambda$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator ${\Lambda }^{\perp }\subseteq \hat{G}$. We investigate the validity of the following statement: For every $\eta$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{M_{\tau }{T}_{\lambda }\eta {\}}_{\lambda \in \Lambda ,\tau \in {\Lambda }^{\perp }}$ is not a frame for $L^2(G)$. When $G=\mathbb{R}$ and $\Lambda =\alpha \mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\Lambda )$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\Lambda )\times (\hat{G}/{\Lambda }^{\perp })$. We prove this equivalence using a connection between Gabor frames and Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert $C^{\ast }$-modules. As an application, we prove a new Balian-Low theorem for the group $\mathbb{R}\times {\mathbb{Q}}_p$, where ${\mathbb{Q}}_p$ denotes the $p$-adic numbers.

29 pages

References in corpus (3)

• Heisenberg modules as function spaces
• Time-frequency analysis on the adeles over the rationals
• Projective multiresolution analyses for dilations in higher dimensions