A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials
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| Format: | Preprint |
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2026
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| _version_ | 1866918305973927936 |
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| author | Cruz-Barroso, Ruymán Fernández, Lidia Marcellán, Francisco |
| author_facet | Cruz-Barroso, Ruymán Fernández, Lidia Marcellán, Francisco |
| contents | A new alternative numerical procedure to the Szegő quadrature formulas for the estimation of integrals with respect to a positive Borel measure $μ$ 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 $μ$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials Cruz-Barroso, Ruymán Fernández, Lidia Marcellán, Francisco Numerical Analysis 33C45, 42C05, 65D32, 41A55, 62J05 A new alternative numerical procedure to the Szegő quadrature formulas for the estimation of integrals with respect to a positive Borel measure $μ$ 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 $μ$, 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. |
| title | A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials |
| topic | Numerical Analysis 33C45, 42C05, 65D32, 41A55, 62J05 |
| url | https://arxiv.org/abs/2601.18721 |