A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials

Abstract

A new alternative numerical procedure to the Szeg\H{o} quadrature formulas for the estimation of integrals with respect to a positive Borel measure $\mu$ supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand $F$ are only known at a finite number of points, which we will assume to be uniformly distributed on the unit circle (although this does not actually constitute a restriction). Our technique consists of obtaining an approximating Laurent polynomial $L$ to $F$ by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to $\mu$, and to use the values of $F$ at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.