论文标题
多边形总和的Euler-Maclaurin公式
An Euler-Maclaurin formula for polygonal sums
论文作者
论文摘要
我们证明了一个双重多边形总和的Euler-Maclaurin公式，作为推论，我们获得了具有整数顶点的多边形的平滑函数积分的近似正交公式。我们的Euler-Maclaurin公式是在Pick的定理上使用谐波分析中的工具，它涉及整数多边形中整数点的数量，并涉及加权的Riemann总和。最后，我们还展示了一个古典技巧，可以追溯到Huygens和Newton，以加速这些Riemann总和的融合。
We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.
