Gabor frames with Hermite functions (Q868964)

We investigate Gabor frames based on a linear combination of Hermite functions \(H_n\). We derive sufficient conditions on the lattice \(\Lambda\subseteq\mathbb R^2\) such that the Gabor system \(\{e^{2\pi i\lambda_2t}H_n(t-\lambda_1)\colon\lambda=(\lambda_1,\lambda_2)\in\Lambda\}\) is a frame. An example supports our conjecture that our conditions are sharp. The main tools are growth estimates for the Weierstrass \(\sigma\)-function and a new type of interpolation problem for entire functions on the Bargmann-Fock space.