伯恩斯坦多项式、Bergman核与环面Kähler簇
Bernstein polynomials, Bergman kernels and toric Kähler varieties
摘要：
以前似乎没有观察到经典的伯恩斯坦多项式$B_N(f)(x)$与$\CP^1$上的Fubini-Study度量的Bergman-Szego核$Π_N$密切相关：$B_N(f)(x)$是Toeplitz算子$Π_N f(N^{-1} D_θ)$的Berezin符号。这种关系表明伯恩斯坦多项式推广到任何复曲面Kahler簇和Delzant多面体$P$。当$f$是光滑的时，$B_N(f)(x)$允许完全的渐近展开。在$P$上积分给出了$N P$中格点上光滑函数的Dedekind-Riemann和的完全渐近展开式，与Guillemin-Sternberg等人的Euler-MacLaurin和公式有关.
英文摘要：
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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