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Gabor (Super)Frames with Hermite Functions

By Karlheinz Gröchenig and Yurii Lyubarskii


Abstract. We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions Hn. Let h = (H0, H1,..., Hn) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R 2 such that the Gabor system {e 2πiλ2t h(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a frame for L 2 (R, C n+1). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames. 1

Year: 2013
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