Some error bounds for Gauss–Jacobi quadrature rules

Abstract

We estimate the error of Gauss–Jacobi quadrature rule applied to a function f, which is supposed locally absolutely continuous in some Besov type spaces, or of bounded variation on [−1,1]. In the first case the error bound concerns the weighted main part φ-modulus of smoothness of f introduced by Z. Ditzian and V. Totik, while in the second case we deal with a Stieltjes integral with respect to f. The stated estimates generalize several error bounds from literature and, in both the cases, they assure the same convergence rate of the error of best polynomial approximation in weighted L1 space.