Gabor frames with rational density

2011/08/12 by Lyubarskii, Yurii, Nes, Preben Gråberg
#33C90 #42C15 #94A12 #FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.1108.2684

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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