Multi-window Gabor systems on discrete periodic sets

TL;DR

This paper investigates multiwindow discrete Gabor systems on discrete periodic sets, providing matrix-conditions for frames, Riesz bases, and orthonormal bases, along with characterizations, duals, perturbations, and K-frames.

Contribution

It offers new matrix-criteria and characterizations for various Gabor system properties on discrete sets, including frames, bases, duals, and K-frames, extending existing theory.

Findings

Matrix-conditions for Gabor frames and bases are established.

Characterizations of dual and perturbation properties are provided.

Construction methods for M-D-G K-frames are introduced.

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.