Multipoint rational approximants with preassigned poles

Abstract

Let $\mu$ be a finite positive Borel measure whose support $S(\mu)$ is a compact regular set contained in $\Bbb R$. For a function of Markov type $\hat\mu(z)=\int_{S(\mu)}d\mu(x)/(z-x)$, $z\in\Bbb C\sbs S(\mu)$, we consider multipoint Padé-type approximants (MPTAs), where some poles are preassigned and interpolation is carried out along a table of points contained in $\overline{\Bbb C}\sbs {\rm Co}(S(\mu))$ which is symmetrical with respect to the real line. The main purpose of this paper is the study of the `exact rate of convergence' of the MPTAs to the function $\hat\mu$.