  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.
