Discrete Zak Transform and Multi-window Gabor Systems on Discrete Periodic Sets

In this paper, $\mathcal{G}(g,L,M,N)$ denotes a $L-$window Gabor system on a periodic set $\mathbb{S}$, where $L,M,M\in \mathbb{N}$ and $g=\{g_l{\}}_{l\in {\mathbb{N}}_L}\subset {\mathrm{\ell }}^2(\mathbb{S})$. We characterize which $g$ generates a complete multi-window Gabor system and a multi-window Gabor frame $\mathcal{G}(g,L,M,N)$ on $\mathbb{S}$ using the Zak transform. Admissibility conditions for a periodic set to admit a complete multi–window Gabor system, multi-window Gabor (Parseval) frame, and multi–window Gabor (orthonormal) basis $\mathcal{G}(g,L,M,N)$ are given with respect to the parameters $L$, $M$ and $N$.