Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

Details Szasz Analytic Functions and Noncompact Kähler Toric Manifolds Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

2008-09-15

arxiv.org/abs/0809.2436v5

Mathematics

Differential Geometry

To appear in Jounal of Geometric Analysis

Scientific paper

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.

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