Gabor orthonormal bases, tiling and periodicity

Abstract

We show that if the Gabor system $\{g(x-t)e^{2\pi isx}\}$, $t\in T$, $s\in S$, 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.

To appear in Mathematische Annalen

References in corpus (1)

• Tiling by translates of a function: results and open problems