Extremal polynomials with preassigned zeros and rational approximants

Let p(n) be the nth orthonormal polynomial with respect to a positive finite measure mu supported by Delta = [-1, 1]. It is well known that, uniformly on compact subsets of C\Delta, [GRAPHICS] and, for a large class of measures mu, [GRAPHICS] where g(Omega)(z) is Green's function of Omega = (C) over bar\Delta with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Pade approximants of the Markov function [GRAPHICS] to f on Omega with a certain geometric speed measured by g(Omega)(z). We prove corresponding results when we restrict the freedom of p(n) by preassigning some of the zeros. This means that the Pade approximants are replaced by Pade-type approximants where some of the poles are preassigned. We also replace Delta by general compact subsets of C.