Gaussian quadrature formulae on the unit circle

Let mu be a probability measure on [0, 2pi]. In this paper we shall be concerned with the estimation of integrals of the form I-mu(f) = (1/2pi) integral(0)(2pi) f(e(i0)) dmu(theta). For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szego polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Pade approximants for the Herglotz-Riesz transform of mu. Furthermore, a comparison with the so-called Szego quadrature formulae is presented through some illustrative numerical examples. (C) 2002 Elsevier Science B.V. All rights reserved.