On the structure of the Gram matrix for Gabor systems generated by B-splines

TL;DR

This paper investigates the Gram matrix structure of Gabor systems generated by B-splines, revealing a block-Toeplitz structure that facilitates spectral analysis of finite submatrices.

Contribution

It demonstrates that certain submatrices of the Gram matrix have a block-Toeplitz structure, enabling spectral analysis for Gabor systems generated by B-splines.

Findings

Submatrices of the Gram matrix exhibit block-Toeplitz structure.

Spectral results for finite sub-blocks are derived using Toeplitz matrix theory.

Application to Gabor systems generated by B-splines of order N.

Abstract

We consider the Gabor system $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ generated by a continuous, compactly supported function $g$ over the time-frequency lattice generated by the parameters $a$ and $b$. We show that, under an appropriate ordering of the Gabor elements, certain submatrices of the Gram matrix of $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ exhibit a block-Toeplitz structure. This structural property enables us to derive spectral results for finite sub-blocks of the Gram matrix by appealing to the spectral theory of Toeplitz matrices. In particular, we apply our results to the Gram matrix of Gabor systems generated by the $N$th-order B-spline.