## Abstract

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^{-1}Dθ) 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.

Original language | English (US) |
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Pages (from-to) | 51-76 |

Number of pages | 26 |

Journal | Journal of Symplectic Geometry |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2009 |

## ASJC Scopus subject areas

- Geometry and Topology