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

Johannes Testorf

TL;DR

The paper develops a geometric framework for the discrete Gabor Transform by embedding time–frequency data on a flat torus and interpreting Bargmann-type images as theta line bundles over Abelian varieties. It establishes higher-dimensional frame and interpolation criteria in terms of algebraic-geometric invariants (multipoint pseudoeffective thresholds and Seshadri constants) and shows that sampling questions reduce to theta-function zeros instead of large holomorphic spaces. The approach unifies DGT frame theory with Toeplitz operator techniques and Bergman-kernel asymptotics, yielding explicit high-dimensional results and asymptotic localization behavior via restriction operators. This bridges time-frequency analysis with complex geometry, enabling robust, dimension-agnostic criteria for frames, interpolation, and localization on tori and Abelian varieties, with potential extensions to more general complex manifolds.

Abstract

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.

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

TL;DR

The paper develops a geometric framework for the discrete Gabor Transform by embedding time–frequency data on a flat torus and interpreting Bargmann-type images as theta line bundles over Abelian varieties. It establishes higher-dimensional frame and interpolation criteria in terms of algebraic-geometric invariants (multipoint pseudoeffective thresholds and Seshadri constants) and shows that sampling questions reduce to theta-function zeros instead of large holomorphic spaces. The approach unifies DGT frame theory with Toeplitz operator techniques and Bergman-kernel asymptotics, yielding explicit high-dimensional results and asymptotic localization behavior via restriction operators. This bridges time-frequency analysis with complex geometry, enabling robust, dimension-agnostic criteria for frames, interpolation, and localization on tori and Abelian varieties, with potential extensions to more general complex manifolds.

Abstract

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.

Paper Structure

This paper contains 13 sections, 22 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. Geometric Preliminaries: Theta Line Bundles, Seshadri Constants, Toeplitz Operators, and Asymptotic Results
  4. Theta Line Bundles
  5. Pseudoeffective Thresholds and Seshadri Constants
  6. Asymptotics of Line bundles
  7. Toeplitz Operators
  8. The Higher Dimensional DGT
  9. Frame results for the DGT
  10. Restriction Operators on $S_N$
  11. Restriction Operators
  12. Asymptotic Results
  13. Terminology of Line Bundles

Key Result

Theorem 1.1

The points $\{(k_j,l_j)\}_{j=1}^{K}\subseteq I_N^2$ induce a Gabor frame for the DGT on $\mathbb C^{N^d}$ with window $h^{\Omega/N}_N$, i.e. for any $\mathbf f\in \mathbb C^{N^d}$ there exists constants $A,B>0$ such that when where $z_j = -i(N^{-1} \Omega k_j+\frac{l_j}{N})\in \mathbb T_\Omega,$ and $\lambda(L,z_1,...,z_K)$ denotes the multipoint pseudoeffective threshold. (See section 2.2) Furt

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5: BerToep
  • ...and 24 more