An extended relation between orthogonal rational functions on the unit circle and the interval [−1,1]

The relation between orthogonal polynomials on the unit circle and the interval has been given by Szegő. In this paper we generalize this relation to orthogonal rational functions with prescribed complex poles. As an application we obtain explicit expressions for the orthogonal rational functions with respect to the weight functions 1±cosθ on the unit circle, respectively {(1+x)/(1−x)}±1/2 on the interval and use one of them to illustrate the relation.