Bernstein polynomials, Bergman kernels and toric Kähler varieties

Details Bernstein polynomials, Bergman kernels and toric Kähler varieties Bernstein polynomials, Bergman kernels and toric Kähler varieties

2007-05-20

arxiv.org/abs/0705.2879v3

J. Sympl. Geom. 7 (2) (2009), 51-76

Mathematics

Complex Variables

Fixed some more typos and added some detail

Scientific paper

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 $N P$ related to Euler-MacLaurin sum formulae of Guillemin-Sternberg and others.

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