Gabor (Super)Frames with Hermite Functions

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 $Λ\subseteq \bR ^2$ such that the Gabor system $\{e^{2πi λ_2 t} \boh (t-λ_1): λ= (λ_1, λ_2) \in Λ\}$ is a frame for $L^2 (\bR, \bC ^{n+1})$. 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 $\sigma$-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames.