Time-Frequency Shift Invariance of Gabor Spaces with an $S_0$-Generator

Abstract

We consider Gabor Riesz sequences generated by a lattice $\Lambda \subset {\mathbb{R}}^2$ and a window function $g\in L^2(\mathbb{R})$ which is well localized in both time and frequency. When $g$ belongs to the Feichtinger algebra, we prove that only those time-frequency shifts with parameters from the lattice $\Lambda$ leave the corresponding Gabor space invariant. This improves on earlier results where only lattices of rational density were considered. A slightly weaker result is proved - again for lattices of general density - under the regularity assumptions of the classical Balian-Low theorem, where both $g$ and its Fourier transform belong to the Sobolev space $H^1(\mathbb{R})$. The proof relies on a combination of methods from time-frequency analysis and the theory of $C^{\ast }$-algebras, specifically the so-called irrational rotation algebra.