Quadrature on the half line and two-point Pade approximants to Stieltjes functions. Part III. The unbounded case

摘要

Let α be a general, absolutely continuous measure, possibly complex, supported on [0,∞). Let Fα(z) denote its Cauchy transform. In this paper we prove, under suitable conditions, the convergence of two-point Padé type approximants to Fα and of the associated quadrature formulas ∑j=1nAjf(xj) to the intergral ∫_0~∞ f(x)da(x). These quadrature formulas can be of Gaussian or of interpolatory type. Estimates for the rate of convergence are also included.