Bernstein measures on convex polytopes
arXiv:0805.3379
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 -simplex obtained by Abel-Ivan and Hörmander. 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.
To appear in a Contemporary Math volume for the proceedings of the conference "Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada's 60th birthday" A mistake in the statement of Proposition 2.8 and its proof are fixed. References are updated