The convergence properties of orthogonal rational functions on the extended real line and analytic on the upper half plane

We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel and reproducing kernel. Those rational measures represent the same inner product as the normalized Nevanlinna measure in [Formula: see text]. Moreover, we consider the convergence properties of the interpolants which interpolate the Carathéodory function [Formula: see text].