Gabor (super)frames with Hermite functions

By Karlheinz Gröchenig, Yurii Lyubarskii. 2008

128 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/ccdc5e4d884d94cbc7dce6b73c12133503dd92d4

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In Applied Mathematics

In Computational Mechanics

In Signal Processing

Citations in the corpus

Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions (2009, methodology): “The orthogonal basis property has the following consequence: Corollary 1.”
On the structure of Gabor and super Gabor spaces (2010, background): “which provides the continuous counterpart of the theory of super Gabor frames presented in [3] and developed in the case of Hermite functions in [6,8, 9 ].”
Gabor Frames and Totally Positive Functions (2011, background): “The construction of a dual window is underlying the proofs that G(g, α, β) is a frame for the Gaussian g(t) = e 2 in [16] and for the one-sided exponential g(t) = eχR+(t) [19].”
The Mystery of Gabor Frames (2014, background): “In [44,45]we proved that for every lattice ⊆ R2 with vol ( ) < 1 n+1 the Gabor familyG(g, ) is a frame.”