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 $${\Lambda \subseteq \mathbb{R} ^2}$$ 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.