Gabor orthonormal bases, tiling and periodicity

TL;DR

This paper proves that Gabor orthonormal bases with compactly supported windows in L^2(R) require periodic time and frequency shift sets, by establishing a functional tiling condition that links tiling and periodicity.

Contribution

It introduces a necessary tiling condition for Gabor bases and demonstrates that both shift sets must be periodic if the window is compactly supported.

Findings

Both T and S are periodic for Gabor orthonormal bases with compact support

Established a tiling condition linking Gabor bases and periodicity

The tiling condition may be useful in other functional analysis contexts

Abstract

We show that if the Gabor system $\{ g(x-t) e^{2\pi i s x}\}$, $t \in T$, $s \in S$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.