Mathematische Annalen

, 345:267 | Cite as

Gabor (super)frames with Hermite functions

Article

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 \({\Lambda \subseteq \mathbb{R} ^2}\) 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.

Mathematics Subject Classification (2000)

42C15 33C90 94A12 

Notes

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Department of MathematicsNorwegian University of Science and TechnologyTrondheimNorway

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