Gaussian Gabor frames, Seshadri constants and generalized BuserSarnak invariants
Abstract
We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander's $\dbar$method, the OhsawaTakegoshi extension theorem and a Kählervariant of the symplectic embedding theorem of McDuffPolterovich for ellipsoids. Our approach is based on the wellknown link between sets of interpolation for the BargmannFock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the BargmannFock space, and give also a condition in terms of the generalized BuserSarnak invariant of the lattice. Our results on Gaussian Gabor frames are in terms of the Sehsadri constant and the generalized BuserSarnak invariant of the associated symplectic dual lattice. The theory of Hörmander estimates and the OhsawaTakegoshi extension theorem allow us to give estimates for the frame bounds in terms of the BuserSarnack invariant and in the onedimensional case these bounds are sharp thanks to Faltings' work on Green functions in Arakelov theory.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.04988
 Bibcode:
 2021arXiv210704988L
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Functional Analysis