[Paper Review] Szasz Analytic Functions and Noncompact Toric Varieties

Renjie Feng|arXiv (Cornell University)|Sep 15, 2008

Geometry and complex manifolds2 citations

TL;DR

This paper establishes a correspondence between Szász analytic functions and the Bergman kernel on the Bargmann-Fock space, generalizing Szász approximation to infinite-volume toric Kähler varieties. It demonstrates that the Szász function arises as the universal scaling limit of Bernstein polynomials near the boundary, enabling summation of lattice points over infinite-volume polytopes.

ABSTRACT

Abstract. We relate the classical approximations SN(f)(x) of O.Szasz to the Bergman kernel of the Bargmann-Fock space H2 (C, e−N|z|2dm(z)). This relation is the analogue for compact toric varieties of the relation between Bernstein polynomials and Bergman kernels on compact toric Kähler varieties of S. Zelditch. The relation is then used to generalize the Szasz analytic functions to any infinite volume toric Kähler variety. Further, we show that the Szsaz analytic function is the universal scaling limit of the Bernstein polynomial as a point approaches the boundary. Applications to summing lattice points over infinite volume polytopes are given. 1.

Motivation & Objective

• To extend the classical Szász approximation to noncompact toric Kähler varieties using Bergman kernel methods.
• To establish a duality between Szász analytic functions and the Bargmann-Fock space's Bergman kernel, analogous to Zelditch's results for compact varieties.
• To derive the universal scaling limit of Bernstein polynomials as a boundary approach occurs, identifying the Szász function as the limiting object.
• To apply the framework to summing lattice points over infinite-volume polytopes, providing a new analytical tool in toric geometry.

Proposed method

• Relates the Szász operator SN(f)(x) to the Bergman kernel of the Bargmann-Fock space H²(C, e−N|z|²dm(z)).
• Uses the Bergman kernel's reproducing property to define a coherent state representation on the complex plane with Gaussian measure.
• Applies asymptotic analysis to show that as a point approaches the boundary of the polytope, the Bernstein polynomial converges universally to the Szász function.
• Generalizes the construction to any infinite-volume toric Kähler variety via symplectic and complex geometry techniques.
• Employs the holomorphic Fock space structure to define Szász analytic functions as limits of polynomial approximations.
• Applies the resulting framework to count lattice points in unbounded polytopes using analytic continuation and integral transforms.

Experimental results

Research questions

• RQ1 How does the Szász approximation operator relate to the Bergman kernel in the Bargmann-Fock space?
• RQ2 Can the Szász analytic function be generalized beyond compact toric varieties to infinite-volume Kähler manifolds?
• RQ3 What is the universal scaling limit of Bernstein polynomials as a point approaches the boundary of a toric variety?
• RQ4 How can the asymptotic behavior of these operators be used to sum lattice points over unbounded polytopes?
• RQ5 What is the geometric and analytic role of the Bergman kernel in extending approximation theory to noncompact settings?

Key findings

• The Szász operator SN(f)(x) is shown to be equivalent to the Bergman kernel reproducing formula in the Bargmann-Fock space H²(C, e−N|z|²dm(z)).
• The Szász analytic function emerges as the universal scaling limit of Bernstein polynomials when the evaluation point approaches the boundary of the polytope.
• The construction generalizes Szász approximation to any infinite-volume toric Kähler variety through the use of the Bergman kernel and coherent states.
• The method enables the summation of lattice points over infinite-volume polytopes via analytic continuation and integral representations.
• The duality between Szász functions and Bergman kernels mirrors Zelditch’s results for compact toric varieties, but extends to noncompact settings.
• The framework provides a new analytical tool for lattice point counting in unbounded convex domains using asymptotic analysis of approximation operators.