Convergence of the Gaussian quadratures for the variable-signed weight function

Abstract

When the weight function ω(x) is allowed to change sign inside the interval of integration, a quadrature formula with respect to the weight function ω(x) was found by G. W. Struble. But the convergence has not been treated. Here we discuss the mean and quadratic mean convergence and the convergence behaviour in the uniform norm for the formula. Also, we give the rate of convergence in terms of the usual modulus of continuity of f, ω(f;h) and an application.