On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture

The frame set conjecture for Hermite functions formulated in [13] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α and modulation rates β that avoid all known obstructions lead t...

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Bibliographic Details
Published inApplied and computational harmonic analysis Vol. 76; p. 101747
Main Authors Horst, Andreas, Lemvig, Jakob, Videbæk, Allan Erlang
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2025
Subjects
Frame
Frame set
Gabor system
Hermite functions
Zak transform
Zeevi matrix
secondary
Frame
Zak transform
Hermite functions
Frame set
Zeevi matrix
Gabor system
primary
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ISSN1063-5203
DOI10.1016/j.acha.2025.101747

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Summary:The frame set conjecture for Hermite functions formulated in [13] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α and modulation rates β that avoid all known obstructions lead to Gabor frames for L2(R). By results in [24,25] and [22], it is known that the conjecture is true for the Gaussian, the 0th order Hermite functions, and false for Hermite functions of order 2,3,6,7,10,11,…, respectively. In this paper we disprove the remaining cases except for the 1st order Hermite function.
ISSN:1063-5203
DOI:10.1016/j.acha.2025.101747