Bernstein measures on convex polytopes

TL;DR

This paper introduces Bernstein measures and approximations on convex polytopes, generalizing classical Bernstein polynomials, and explores their properties, asymptotic behavior, and relations to other approximation methods.

Contribution

It generalizes Bernstein polynomials to convex polytopes and analyzes their properties and asymptotic expansions, extending classical approximation results.

Findings

Defined Bernstein measures on convex polytopes

Proved asymptotic expansion for smooth functions

Compared with Zelditch's Bergman-Bernstein approximations

Abstract

We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and H\"{o}rmander. These are different from the Bergman-Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations.