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

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