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

TL;DR

This work develops a Zak-transform framework to characterize when multi-window discrete Gabor systems on $N\mathbb{Z}$-periodic sets $\mathbb{S}$ form complete systems or frames in $\ell^2(\mathbb{S})$. By introducing a matrix-valued Zak function $Z_g$ and analyzing its rank in the Zak domain, the authors derive precise rank conditions and operator inequalities that determine completeness, frame bounds, and basis properties in terms of $L,M,N$ and the combinatorial counts $|\mathcal{K}_j|$ and $|\mathbb{S}_N|$. They provide admissibility criteria and constructive schemes using disjoint index sets $E_l$ to realize Parseval/tight frames and to identify when Riesz or orthonormal bases are achievable, tying these properties to $|\mathcal{K}_j|\le qL$ and $|\mathbb{S}_N|=LM$. The results unify density-type conditions with Zak-domain algebra for discrete Gabor analysis on periodic sets and include concrete examples illustrating the multi-window constructions.

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 $N$.

Theorems & Definitions (44)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 1
• Lemma 6
• Lemma 7
• Lemma 8
• proof