Szasz Analytic Functions and Noncompact K\"{a}hler Toric Manifolds

TL;DR

This paper extends the classical Szasz analytic function to noncompact Kähler toric manifolds, demonstrating its asymptotic behavior and universal scaling limits, with explicit examples including the unit ball.

Contribution

It generalizes the Szasz analytic function to a broad class of noncompact Kähler manifolds and establishes its asymptotic properties and scaling limits.

Findings

Szasz analytic function is derived from Bergman kernels via pseudo-differential operators.

The generalized Szasz function admits complete asymptotics on noncompact toric manifolds.

Explicit computation of the generalized Szasz function for the Bergman metric on the unit ball.

Abstract

We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_{\theta})$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized Szasz analytic function $S_{h^N}(f)(x)$. About $S_{h^N}(f)(x)$, we prove that it admits complete asymptotics and there exists a universal scaling limit. % We also consider some dilation operator composed with $S_{h^N}(f)(x)$ and we give an estimate about this composition. As an example, we will further compute $S_{h^N}(f)(x)$ for the Bergman metric on the unit ball.