Quadrature on the half-line and two-point Pade approximants to Stieltjes functions—II: Convergence

摘要

Let a be a distribution function on [a,b] (0 ≤ a < b ≤ +∞) such that the moment ck = bad(x) exist for all the integers k. The main course of the paper is to give convergence results both for sequences of two-point Pade approximants to the Cauchy transform of the distribution (Stieltjes function) and for quadrature formulas based on exactly integrating Laurent polynomials. The case when d(x) = k(x)dx, k(x) being a possibly complex function is also considered and estimates of the rate of convergence are given.