Gaussian quadrature formulae on the unit circle

摘要

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form Iμ(f)=(1/2π)∫02πf(eiθ)dμ(θ). For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö 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 Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.