Stable high-order quadrature rules with equidistant points

Newton-Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function values are only available on a regular grid. Yet, these rules rapidly become unstable for high orders. In this paper we review two techniques to construct stable high-order quadrature rules using equidistant quadrature points. The stability follows from the fact that all coefficients are positive. This result can be achieved by allowing the number of quadrature points to be larger than the polynomial order of accuracy. The computed approximations then implicitly correspond to the integral of a least squares approximation of the integrand. We show how the underlying discrete least squares approximation can be optimised for the purpose of numerical integration.