Bernstein measures on convex polytopes

Details Bernstein measures on convex polytopes Bernstein measures on convex polytopes

2008-05-22

arxiv.org/abs/0805.3379v3

Mathematics

Functional Analysis

To appear in a Contemporary Math volume for the proceedings of the conference "Spectral Analysis in Geometry and Number Theory

Scientific paper

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.

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