On Schauder Bases Properties of Multiply Generated Gabor Systems

TL;DR

This paper characterizes when certain Gabor systems, generated by finite sets in L^2(R), form Schauder bases, using a Muckenhoupt matrix condition on an associated Zibulski-Zeevi matrix.

Contribution

It provides a new characterization of Schauder basis properties for multiply generated Gabor systems via a Muckenhoupt matrix condition.

Findings

Schauder basis properties are characterized by a matrix condition.

The results apply to Gabor systems with rational time-frequency shifts.

A specific ordering on the index set is used for the characterization.

Abstract

Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a specific ordering on $\mathbb{Z}\times \mathbb{Z}\times A$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.