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

TL;DR

This paper proves that Gabor spaces generated by well-localized functions are invariant only under lattice shifts, extending previous results to arbitrary lattice densities using advanced time-frequency and algebraic methods.

Contribution

It establishes the invariance properties of Gabor spaces for general lattices with well-localized generators, improving upon prior rational-density restrictions.

Findings

Invariance under lattice shifts for generators in the Feichtinger algebra.

Extension of results to arbitrary lattice densities.

Application of $C^*$-algebra techniques to time-frequency analysis.

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.