Bernstein Polynomials, Bergman Kernels and Toric Kähler Varieties

We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN(f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler variety and Delzant polytope, and gives a novel definition of Bernstein ‘polynomials’ BhN (f) relative to any toric Kähler variety. They uniformly approximate any continuous function f on the associated polytope P with all the properties of classical Bernstein polynomials. Upon integration over the polytope one obtains a complete asymptotic expansion for the Dedekind-Riemann sums 1 Nm ∑ α∈NP f( α N ) of f ∈ C∞(Rm), of a type similar to the Euler-MacLaurin formulae.