Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

Abstract

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)⁻¹dμ(t)<∞. In this paper we study the order of the uniform approximation of the function

$$\widehat\mu = \smallint \tfrac{{d\mu (t)}}{{t - z}}, z \in \mathbb{C},$$

on the disk |z|≤1 and on the closed interval [−1, 1] by means of the orthogonal projection of$\widehat\mu$ on the set of rational functions of degreen. Moreover, the poles of the rational functions are chosen depending on the measure μ. For example, it is shown that if supp μ is compact and does not contain 1, then this approximation method is of best order. But if supp μ=[1,a],a>1, the measure μ is absolutely continuous with respect to the Lebesgue measure, and$\mu '\left( t \right) _\frown ^\smile \left( {t - 1} \right)^\alpha$ fort∈[1,a] and some α>0, then the order of such an approximation differs from the best only by$\sqrt n$.