Orthogonal rational functions and quadrature on the unit circle

In this paper we shall be concerned with the problem of approximating the integral I_µ{f} = ∫_{-π}^π f(e^{it})dµ(t), by means of the formula I_n{f} = ∑_{j=1,...,n} A_j^{(n)} f(x_j^{(n)}) where µ is some finite positive measure. We want the approximation to be so that I_n{f} = I_µ{f} for f 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 weights A_j^{(n)}, it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szegö polynomials.status: publishe