Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants

Abstract

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander's ∂-method, the Ohsawa-Takegoshi extension theorem and a Kähler-variant of the symplectic embedding theorem of McDuff-Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of the Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. Our results on Gaussian Gabor frames are in terms of the Sehsadri constant and the generalized Buser-Sarnak invariant of the associated symplectic dual lattice. The theory of Hörmander estimates and the Ohsawa-Takegoshi extension theorem allow us to give estimates for the frame bounds in terms of the Buser-Sarnack invariant and in the one-dimensional case these bounds are sharp thanks to Falting's work on Green functions in Arakelov theory.