Gaussian Summation: An Exponentially Converging Summation Scheme

Abstract

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.