Bernstein polynomials, bergman kernels and toric Kähler varieties

We show that the classical Bernstein polynomials B_N(f)(x) on the interval [0, 1] (and their higher dimensional eneralizations on the simplex Σ_m C ℝ^m) may be expressed in terms of Bergman kernels forthe Fubini-Study metric on ℂP^m : B_N(f)(x) is obtained by applyingthe 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"B_h^N (f) relative to any toric Kähler variety. They uniformly approximate any continuous function f on the associated polytope Pwith all the properties of classical Bernstein polynomials. Upon integration over the polytope, one obtains a complete asymptotic expansionfor the Dedekind-Riemann sums 1/N^m ΣαaεNP f( α/N ) of f ε C∞(ℝ^m), of a type similar to the Euler-MacLaurin formulae.