Admissibility of Multi-window Gabor Systems in Periodically Supported $\ell^2$-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 $\ell^2(\mathbb{S}, \mathbb{C}^R)$ 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 $g = \{g_l\}_{l \in \mathbb{N}_L} \subset \ell^2(\mathbb{S}, \mathbb{C}^R)$. 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 $\ell^2(\mathbb{S}, \mathbb{C}^R)$. 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 $R$.