On the convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf¹_–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp_n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].