Gabor (super)frames with Hermite functions

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 subset of R-2 such that the Gabor system {e(2 pi i lambda 2t)h(t - lambda(1)) : lambda = (lambda(1), lambda(2)) is an element of Lambda} is a frame for L-2(R, Cn+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.