Multi-window Gabor systems on discrete periodic sets

TL;DR

This work develops a comprehensive matrix-analytic framework for multi-window discrete Gabor frames on discrete periodic sets. It derives necessary and sufficient matrix criteria, via the symbols $\mathcal{M}_{g_l}(j)$ and the frame operator, to determine when $\mathcal{G}(g,L,M,N)$ forms a frame, a Parseval frame, a Riesz basis, or an orthonormal basis in $\ell^2(\mathbb{S})$ or $\ell^2(\mathbb{Z})$, and it characterizes dual frames and perturbations. The discrete Zak transform is employed to obtain complete-system and frame characterizations in the case $M=N$, with explicit conditions for Riesz/orthonormal bases when $N=LM$. The paper then generalizes to $K$-frames, providing practical sufficient matrix-conditions for $\mathcal{G}(g,L,M,N)$ to be a $K$-frame and offering construction methods for $K$-frames that are not ordinary frames, along with a discussion of $K$-duals and minimality. Collectively, the results give a robust, implementable set of tools for designing and analyzing M-D-G frames and their duals on discrete sets with potential applications in digital signal processing and time-frequency analysis.

Abstract

In this paper, we study multiwindow discrete Gabor $(M-D-G)$ systems $\mathcal{G}(g,L,M,N)$ on discrete periodic sets $\mathbb{S}$ and give some necessary and/or sufficient matrix-conditions for a $M-D-G$ system in $\ell^2(\mathbb{S})$ to be a frame. We characterize, also, which $M-D-G$ frames are Riesz bases by the parameters $L$, $M$ and $N$. Matrix-characterizations of $M-D-G$ Parseval frames and $M-D-G$ orthonormal bases are also given. Then, we characterize the existence of $M-D-G$ frames, $M-D-G$ Parseval frames, $M-D-G$ Riesz bases and $M-D-G$ orthonormal bases for $\ell^2(\mathbb{Z})$ by the parameters $M$, $N$ and $L$. We present, also, a matrix-characterization of dual $M-D-G$ frames in $\ell^2(\mathbb{S})$. A perturbation matrix-condition of $M-D-G$ frames is also prsented. We, then, show that a pair of $M-D-G$ Bessel systems can generate pairs of M-D-G dual frames. By the Zak-transform, characterizations of complete M-D-G systems and M-D-G frames in $\ell^2(\mathbb{Z})$ are given in the case of $M=N$ and necessary conditions for a M-D-G system to be a Riesz basis/ orthonormal basis for $\ell^2(\mathbb{Z})$ are also given. We, also, study M-D-G $K$-frames in $\ell^2(\mathbb{S})$, where $K\in B(\ell^2(\mathbb{S})\,)$, and presente some sufficient matrix-conditions for a M-D-G system to form a K-frame and give a construction method of M-D-G $K$-frames which are not M-D-G frames and some examples.

Theorems & Definitions (90)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Remark 1