On the finite linear independence of lattice Gabor systems

TL;DR

This paper provides an alternative proof for the finite linear independence of lattice Gabor systems in $L^2( eal^d)$, using spectral theory, and extends Linnell's results to general lattices in one dimension.

Contribution

It introduces a new proof approach based on spectral theory and extends Linnell's theorem to broader lattice settings in one dimension.

Findings

Proof based on spectral theory of random Schrödinger operators

Full strength of Linnell's result for one-dimensional lattices

Extension to general lattices in one dimension

Abstract

In the restricted setting of product phase space lattices, we give an alternate proof of P. Linnell's theorem on the finite linear independence of lattice Gabor systems in $L^2(\mathbb R^d)$. Our proof is based on a simple argument from the spectral theory of random Schr\"odinger operators; in the one-dimensional setting, we recover the full strength of Linnell's result for general lattices.