高斯求和：一种指数收敛的求和方案
Gaussian Summation: An Exponentially Converging Summation Scheme
AI总结：
本文提出高斯求和公式，通过显式构造权重和横坐标，实现对无限级数的指数收敛求和，并应用于Hardy-Littlewood和等问题。
摘要：
高斯求积是数值积分中众所周知的技术。近年来，针对对应有限和的高散测度的高斯求积引起了新的兴趣。在本文中，我们将这些思想推广到一般的无限和，并给出了高斯求和公式的权重和横坐标的显式构造。高斯求和的横坐标具有一个非常有趣的渐近分布函数，带有尖点奇异性。我们将高斯求和技巧应用于文献中讨论的两个问题。我们发现，对于大范围参数，高斯求和对于Hardy-Littlewood和具有极快的收敛速度。对于光滑但具有大尺度a的函数，高斯求和的误差作为求和点数的函数呈指数收敛。高斯求和达到给定精度所需的点数与大尺度的平方根成正比，而其他求和方案至少需要与尺度成正比的函数评估次数。
英文摘要：
Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of GAUSSIAN SUMMATION formulas. The abscissae of the Gaussian summation have a very interesting asymptotic distribution function with a (cusp) singularity. We apply the Gaussian summation technique to two problems which have been discussed in the literature. We find that the Gaussian summation has an extremely rapid convergence rate for the Hardy-Littlewood sum for a large range of parameters. For functions which are smooth but have a large scale, a, the error of Gaussian Summation shows exponential convergence as a function of summation points. The Gaussian summation achieves a given accuracy with a number of points proportional to the sqrt of the large scale whereas other summation schemes require at least a number of function evaluations proportional to the scale.
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