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

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 on the disk |z|≤1 and on the closed interval −1, 1] by means of the orthogonal projection of 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 fort∈1,a] and some α>0, then the order of such an approximation differs from the best only by . Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 362–368, March, 1999.