, Volume 345, Issue 2, pp 267-286
Date: 31 Mar 2009

Gabor (super)frames with Hermite functions

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Abstract

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H n . Let h = (H 0, H 1, . . . , H n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices \({\Lambda \subseteq \mathbb{R} ^2}\) such that the Gabor system \({\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}\) is a frame for \({L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}\) . 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.

K. Gröchenig was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.
Yu. Lyubarskii was partially supported by the Research Council of Norway grant 10323200. This research is part of the European Science Foundation Networking Programme “Harmonic and Complex Analysis”.