Orthogonal rational functions and modified approximants

AbstractLet {αn|n∞ be a sequence in the open unit disk in the complex plane and let $$(\overline {\alpha _k } |\alpha _k | = - 1$$ when αk=0. Let μ be a positive Borel measure on the unit circle, and let {φn}n∞ be the orthonormal sequence obtained by orthonormalization of the sequence {Bn}n∞ with respect to μ. Let {ψn}n∞ be the sequence of associated rational functions. Using the functions φn, ψn and certain conjugates of them, we obtain modified Padé-type approximants to the function $$F\mu (z) = \int\limits_{ - \pi }^\pi {\frac{{t + z}}{{t - z}}} d\mu (\theta ), (t = e^{i\theta } ).$$