Gabor frames in $\ell^2(\mathbf Z)$ and linear dependence

TL;DR

This paper proves that overcomplete Gabor frames in the discrete space $ ext{ell}^2( extbf{Z})$ generated by finitely supported sequences are always linearly dependent, revealing fundamental properties of such frames.

Contribution

It establishes that all overcomplete Gabor frames with finitely supported generators in $ ext{ell}^2( extbf{Z})$ are necessarily linearly dependent, extending to a general setting.

Findings

Overcomplete Gabor frames in $ ext{ell}^2( extbf{Z})$ are linearly dependent.

Dependence holds for frames generated by finitely supported sequences.

Results apply to Gabor systems with specific modulation and translation parameters.

Abstract

We prove that an overcomplete Gabor frame in $ \ell^2(\mathbf Z)$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in $\ell^2(\mathbf Z)$ with modulation parameter $1/M$ and translation parameter $N$ for some $M,N\in \mathbf N,$ and generated by a finite sequence $g$ in $\ell^2(\mathbf Z)$ with $K$ nonzero entries.