Bernstein polynomials, Bergman kernels and toric K\"ahler varieties

TL;DR

This paper reveals a connection between Bernstein polynomials and Bergman kernels on toric Kähler varieties, leading to new asymptotic expansions for sums over lattice points and generalizations of classical approximation tools.

Contribution

It uncovers a novel relationship between Bernstein polynomials and Bergman kernels, extending classical approximation methods to toric Kähler varieties and deriving comprehensive asymptotic expansions.

Findings

Bernstein polynomials are related to Bergman kernels via Berezin symbols.

Asymptotic expansions are obtained for smooth functions on toric varieties.

New formulas generalize Euler-MacLaurin summation for lattice points in polytopes.

Abstract

It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $\Pi_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $\Pi_N f(N^{-1} D_{\theta})$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over $P$ gives a complete asymptotic expansion for Dedekind-Riemann sums of smooth functions over lattice points in $N P$ related to Euler-MacLaurin sum formulae of Guillemin-Sternberg and others.