Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

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.

To appear in Jounal of Geometric Analysis

References in corpus (3)

• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry
• Non-Compact Symplectic Toric Manifolds
• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Cited by in corpus (1)

• Asymptotic Euler-Maclaurin formula over lattice polytopes