On a problem proposed by H. Bra� concerning the remainder term in quadrature for convex functions

Abstract

We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw?1, is of best possible orderO(n ?2) for every bounded convex function. This result solves an open problem proposed by H. Braβ and published in the problem section of the proceedings of the `2. Conference on Numerical Integration' held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hammerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw ?, β(x)=(1?x)?(1+x)β, we show that the above result for Gaussian quadrature is not valid precisely ifw ?, β is unbounded.