  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.
