Bernstein polynomials, Bergman kernels and toric Kähler varieties

Abstract

It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels ${\Pi }_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator ${\Pi }_{N}f(N^{-1}{D}_{\theta })$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over $P$ gives a complete asymptotic expansion for Dedekind-Riemann sums of smooth functions over lattice points in $NP$ related to Euler-MacLaurin sum formulae of Guillemin-Sternberg and others.

Fixed some more typos and added some detail

References in corpus (2)

• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• The Ehrhart Function for Symbols

Cited by in corpus (1)

• Test configurations, large deviations and geodesic rays on toric varieties