Admissibility of Multi-window Gabor Systems in Periodically Supported ${\mathrm{\ell }}^2$-spaces with Vector-valued Sequences

In this paper, $L,M,N,R$ are positive integers, and $\mathbb{S}$ is an $N$-periodic subset of $\mathbb{Z}$. The space ${\mathrm{\ell }}^2(\mathbb{S},{\mathbb{C}}^R)$ denotes the Hilbert space of vector-valued square-summable sequences over $\mathbb{S}$, with values in the complex Euclidean space ${\mathbb{C}}^R$. We consider the (multi-window) Gabor system $\mathcal{G}(g,L,M,N,R)$, generated by applying translations with parameter $nN$, $n\in \mathbb{Z}$, and modulations with parameter $\frac{m}{M}$, $m\in {\mathbb{N}}_M$, to a collection of sequences $g=\{g_l{\}}_{l\in {\mathbb{N}}_L}\subset {\mathrm{\ell }}^2(\mathbb{S},{\mathbb{C}}^R)$. Using the vector-valued Zak transform, we characterize the class of sequences $g$, called windows, that generate a complete Gabor system or a Gabor frame in ${\mathrm{\ell }}^2(\mathbb{S},{\mathbb{C}}^R)$. Furthermore, we provide admissibility conditions under which the periodic set $\mathbb{S}$ supports a complete Gabor system, a Parseval Gabor frame, or an orthonormal Gabor basis, expressed in terms of the parameters $L$, $M$, $N$, and $R$.