Orthogonal rational functions and quadrature on the unit circle

In this paper we shall be concerned with the problem of approximating the integralI {f}= _– f(eⁱ) d(), by means of the formulaI _n {f}= _j=1 ⁿ A _j ⁽ⁿ⁾ f(x _j ⁽ⁿ⁾ ) where is some finite positive measure. We want the approximation to be so thatI _n{f}=I {f} forf belonging to certain classes of rational functions with prescribed poles which generalize in a certain sense the space of polynomials. In order to get nodes {x _j ⁽ⁿ⁾ } of modulus 1 and positive weightsA _j ⁽ⁿ⁾ , it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szeg polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research.