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

Abstract: 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 ).