Orthogonal rational functions and modified approximants

Abstract

Let {α 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 } ∞ be the orthonormal sequence obtained by orthonormalization of the sequence {B n } ∞ with respect to μ. Let {ψ n } ∞ be the sequence of associated rational functions. Using the functions φ n , ψ n and certain conjugates of them, we obtain modified Pade-type approximants to the function $$F\mu (z) = \int\limits_{ - \pi }^\pi {\frac{{t + z}}{{t - z}}} d\mu (\theta ), (t = e^{i\theta } ).$$