A characterization of Gabor Riesz bases with separable time-frequency shifts

TL;DR

This paper fully characterizes Riesz Gabor systems generated by characteristic functions over tiling sets, extending previous results on Gabor bases and establishing conditions for windows to form Riesz bases.

Contribution

It provides a complete characterization of Riesz Gabor systems with characteristic functions over tiling sets and links the properties of these sets to Gabor frame conditions.

Findings

Characterization of Riesz Gabor systems with characteristic functions over tiling sets

Necessary conditions for multi-tiling sets to serve as windows for Riesz Gabor bases

Development of new results on zeros of the Zak transform related to Gabor frames

Abstract

A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $\Lambda\times \Gamma \subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, \Lambda\times \Gamma) = \left\{ e^{2\pi i \xi\cdot t}g(t-x)\right\}_{ (x,\xi)\in \Lambda\times \Gamma }$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=\chi_\Omega$ has been solved by Han and Wang. In this paper, we build on these results and obtain a full characterization of Riesz Gabor systems of the form $\mathcal G(\chi_\Omega, \Lambda\times \Gamma)$ when $\Omega$ is a tiling of $\mathbb{R}^d$ with respect to $\Lambda$. Furthermore, for a certain class of lattices $\Lambda\times \Gamma$, we prove that a necessary condition for the characteristic function of a multi-tiling set to serve as a window function for a Riesz Gabor basis is that the set must be a tiling set. To prove this, we develop new results on the zeros of the Zak transform and connect these results to Gabor frames.