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 to Gabor frames for L 2 ( 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.