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 \( R \).