Gabor (Super)Frames with Hermite Functions

TL;DR

This paper characterizes all lattices that generate Gabor frames using Hermite functions, providing new conditions and bounds, and introduces advanced mathematical tools for analyzing vector-valued Gabor systems.

Contribution

It offers a complete characterization of lattices for vector-valued Gabor frames with Hermite functions and introduces new estimates and conditions for frame generation.

Findings

Complete lattice characterization for Gabor frames with Hermite functions

New sufficient conditions for single Hermite functions to generate frames

Improved lower frame bound estimates

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 $\Lambda \subseteq \bR ^2$ such that the Gabor system $\{e^{2\pi i \lambda_2 t} \boh (t-\lambda_1): \lambda = (\lambda_1, \lambda_2) \in \Lambda \}$ 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.