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

Author: Tatsuya Tate


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