Gabor systems with Hermite functions of order n and oversampling greater than n+1 which are not frames

TL;DR

This paper investigates the limitations of density conditions for Gabor systems with Hermite functions, revealing that oversampling beyond a certain point does not guarantee frame properties, based on Zak transform zeros.

Contribution

It demonstrates that a known density condition for Gabor frames with Hermite functions is not sufficient, highlighting the role of Zak transform zeros in frame analysis.

Findings

Density condition is not sufficient for Gabor frames with Hermite functions.

Zeros of the Zak transform influence the frame property.

Oversampling greater than n+1 does not ensure a frame.

Abstract

We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.