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

Abstract

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 $L^2\left(R\right)$. 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 $2,3,6,7,10,11,\dots$, respectively. In this paper we disprove the remaining cases except for the 1st order Hermite function.