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arXiv · 2311.01547

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

Andreas Horst·Jakob Lemvig·Allan Erlang Videbaek
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Abstract

The frame set conjecture for Hermite functions formulated in [Gröchenig, J. Fourier Anal. Appl., 20(4):865-895, 2014] 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 $L^{2}(\mathbb{R})$. By results in [Seip and Wallstén, J. Reine Angew. Math., 429:107-113, 1992] and [Lemvig, Monatsh. Math., 182(4):899-912, 2017], it is known that the conjecture is true for the Gaussian, the $0$th order Hermite functions, and false for Hermite functions of order $2,3,6,7,10,11,\dots$, respectively. In this paper we disprove the remaining cases except for the $1$st order Hermite function.

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Andreas Horst, Jakob Lemvig, Allan Erlang Videbaek. 2025-06-23. On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture. https://doi.org/10.1016/j.acha.2025.101747

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