September 12, 2025Open Access

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

Key Points

The results show that certain density conditions do not guarantee the frame property in Gabor systems.
A key finding illustrates how the Zak transform's zeros influence the frame property of oversampled systems.
This research addresses limitations of existing conditions, offering new insights into Gabor systems.
The implications suggest that current criteria for frame properties should be re-evaluated for oversampled Gabor systems.

Abstract

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.

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Cite This Study

Markus Faulhuber (Tue,) studied this question.

synapsesocial.com/papers/68d44f7b31b076d99fa56be8 https://doi.org/https://doi.org/10.1007/s43670-025-00112-5