Rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle

In this paper, multipoint rational approximants to the Riesz-Herglotz transform of a Borel measure μ, supported on [−π, π] are considered. We give estimates for the rate of convergence of these approximants. These estimates are obtained from the asymptotic behaviour of a sequence of rational functions, orthogonal with respect to μ and having prescribed poles 1αk, where {αk}1∞ is a sequence of complex numbers compactly contained in the open unit disk. As an application we also give the rate of convergence of rational Szegő quadrature formulas for integrals with respect to μ.