Bernstein polynomials, Bergman kernels and toric K\"ahler 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 ${\mathrm{\Pi }}_N$ for the Fubini-Study metric on ${\text{CP}}^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator ${\mathrm{\Pi }}_{N}f(N^{-1}{D}_{\theta })$. 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.