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 supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand 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 to by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to , and to use the values of at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.
Cite
@article{arxiv.2601.18721,
title = {A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials},
author = {Ruymán Cruz-Barroso and Lidia Fernández and Francisco Marcellán},
journal= {arXiv preprint arXiv:2601.18721},
year = {2026}
}