We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional eneralizations on the simplex Σm C ℝm) may be expressed in terms of Bergman kernels forthe Fubini-Study metric on ℂPm : BN(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"BhN (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/Nm ΣαaεNP f( α/N ) of f ε C∞(ℝm), of a type similar to the Euler-MacLaurin formulae.
ASJC Scopus subject areas
- Geometry and Topology