  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.
