Orthogonal rational functions and quadrature on the unit circle

Abstract

In this paper we shall be concerned with the problem of approximating the integralI μ{f}=∫ π−π f(eiθ) dμ(θ), by means of the formulaI n {f}=Σ n j=1 A (n) j f(x (n) j ) where μ is some finite positive measure. We want the approximation to be so thatI n{f}=I μ{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 (n)j } of modulus 1 and positive weightsA (n)j , it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szegő polynomials.