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

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 نشر من قبل Xu Wang
 تاريخ النشر 2021
  مجال البحث
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We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kahler geometry such as Hormanders $dbar$-method, the Ohsawa--Takegoshi extension theorem and a Kahler-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 a certain extremal type 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 Hormander 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 Faltings work on Green functions in Arakelov theory.



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