Orthogonal rational functions and modified approximants

Abstract

Let {α _n | ^∞ _n be a sequence in the open unit disk in the complex plane and let

$(\overline {\alpha _k } |\alpha _k | = - 1$ when α _k =0. Let μ be a positive Borel measure on the unit circle, and let {φ _n } ^∞ _n be the orthonormal sequence obtained by orthonormalization of the sequence {B _n } ^∞ _n with respect to μ. Let {ψ _n } ^∞ _n be the sequence of associated rational functions. Using the functions φ _n , ψ _n and certain conjugates of them, we obtain modified Padé-type approximants to the function

$$F\mu (z) = \int\limits_{ - \pi }^\pi {\frac{{t + z}}{{t - z}}} d\mu (\theta ), (t = e^{i\theta } ).$$