Gauss Quadratures: An Inverse Problem

Abstract

The problem of finding the Gauss quadrature (i.e., the quadrature formula of the maximal polynomial order) for a given weight function is reversed: a weight function, for which a given quadrature formula with positive weights is the Gauss quadrature, is sought. Among all such weight functions, those minimizing, in the least square sense, the kth derivative ($k > 0$ given) are characterized. An algorithm for calculating the values of the minimizing weight functions is derived and applied to find positive weights for quadratures with equidistant knots. The concepts are further generalized to include Gauss–Radau and Gauss–Lobatto quadratures.