Moments in quadrature problems

An account is given of the role played by moments and modified moments in the construction of quadrature rules, specifically weighted Newton-Cotes and Gaussian rules. Fast and slow Lagrange interpolation algorithms, combined with Gaussian quadrature, as well as linear algebra methods based on moment equations, are described for generating Newton-Cotes formulae. The weaknesses and strengths of these methods are illustrated in concrete examples involving weight functions with and without singularities. New conjectures are formulated concerning the positivity of certain Newton-Cotes formulae for Jacobi weight functions and for the logistics weight, with numerical evidence being provided to support them. Finally, an inherent limitation is pointed out in the use of moment information to construct Gauss-type quadrature rules for the Hermite weight function on bounded or half-infinite intervals.