On Schauder basis properties of multiply generated Gabor systems

Abstract

Let $\mathcal{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 \[ G(1,p/q,\mathcal{A} )=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z} , g\in \mathcal{A} \}, \] with a specific ordering on $\mathbb{Z} \times \mathbb{Z} \times \mathcal{A} $. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.