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

TL;DR

This paper characterizes the conditions under which multi-window Gabor systems on discrete periodic sets form complete systems, frames, or bases using the Zak transform, extending Gabor analysis to periodic discrete structures.

Contribution

It introduces a characterization of multi-window Gabor systems on periodic sets via the Zak transform, including admissibility conditions for completeness, frames, and bases.

Findings

Provides criteria for Gabor system completeness on periodic sets

Establishes conditions for Gabor frames and bases on discrete periodic structures

Uses Zak transform to analyze multi-window Gabor systems

Abstract

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 $g=\{g_l\}_{l\in \mathbb{N}_L}\subset \ell^2(\mathbb{S})$. 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 $N$.