A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann spaces can be expressed as theta line bundles on Abelian varieties. We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get results which hold in higher dimension. We also give an application to asymptotics of restriction operators which arises from the asymptotic behavior of Bergman kernels for high tensor powers.