Gabor frames with rational density

We consider the frame property of the Gabor system G(g, α, β) = {e2πiβnt g(t - αm) : m, n \in Z} for the case of rational oversampling, i.e. α, β \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, α, β) does not generate a frame if αβ = (n-1)/n. Special attention is paid to the first Hermite function h_1(t) = te^(-πt^2).

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