Gabor frames with rational density
Abstract
We consider the frame property of the Gabor system G(g, α, β) = e2πiβnt g(t - αm) : m, n ∈ Z for the case of rational oversampling, i.e. α, β ∈ 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 h1(t) = te(-πt2).
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