Numerical methods for strongly oscillatory and singular functions are given in this paper. Beside a summary of standard methods and product integration rules, we consider a class of complex integration methods, as well as Gaussian quadratures with respect to the oscillatory weight w(x) = xe, x ∈ [−1, 1]. Numerical examples are included.

In analogy to the subject of Gaussian integration formulas we present an overview of some Gaussian summation formulas. The derivation involve polynomials that are orthogonal under discrete inner products and the resulting formulas are useful as a numerical device for summing fairly general series. Several illuminating examples are provided in order to present various aspects of this not very we...

A Chebyshev-type quadrature for a probability measure σ is a distribution which is uniform on n points and has the same first k moments as σ. We give bounds for the smallest possible n required to achieve a certain degree k. In contrast to previous results of this type, our bounds use only simple properties of σ and are thus applicable in wide generality. In particular, it is shown that wheneve...