Bernstein polynomials, Bergman kernels and toric Kähler varieties

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 $Π_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $Π_N f(N^{-1} D_θ)$. 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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