On some Hermite series identities and their applications to Gabor analysis

TL;DR

This paper derives new infinite series identities for Hermite functions, disproves a conjecture about Gabor frames for certain Hermite functions, and suggests the complexity of their Gabor frame sets.

Contribution

It introduces novel Hermite series identities and uses them to disprove the Gabor frame set conjecture for specific Hermite functions and related eigenfunctions.

Findings

Disproved the Gabor frame set conjecture for Hermite functions of order 4m+2 and 4m+3.

Identified that the Gabor frame set for these functions has a complex structure.

Results extend to a broad class of eigenfunctions of the Fourier transform.

Abstract

We prove some infinite series identities for the Hermite functions. From these identities we disprove the Gabor frame set conjecture for Hermite functions of order $4m+2$ and $4m+3$ for $m \in \{0\} \cup \mathbb{N}$. The results hold not only for Hermite functions, but for two large classes of eigenfunctions of the Fourier transform associated with the eigenvalues $-1$ and $i$, and the results indicate that the Gabor frame set of all such functions must have a rather complicated structure.