On Schauder Bases Properties of Multiply Generated Gabor Systems

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 $$G(1,p\mathrm{/}q,A)=\{e^{2\pi imx}g(x-np\mathrm{/}q):m,n\in \mathbb{Z},g\in A\},$$ 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.