Gabor frames with rational density

Abstract: We consider the frame property of the Gabor system G(g, {\alpha}, {\beta}) = {e2{\pi}i{\beta}nt g(t - {\alpha}m) : m, n \in Z} for the case of rational oversampling, i.e. {\alpha}, {\beta} \in Q. A 'rational' analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, {\alpha}, {\beta}) does not generate a frame if {\alpha}{\beta} = (n-1)/n. Special attention is paid to the first Hermite function h_1(t) = te^-{\i}t^2).