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

TL;DR

This work addresses numerical integration on the unit circle with respect to a positive Borel measure $μ$ when the integrand $F$ is known only at a finite set of points. It introduces a mixed interpolation-regression approach that constructs a Laurent polynomial $L$ in a chosen subspace, interpolates at $m$ points near the zeros of a para-orthogonal polynomial, and uses the remaining nodes in a complex regression to refine $F$; Hermite interpolation is employed to enlarge the interpolant’s capability. The method yields an explicit $L$ of the form $L(z)=P_m(z)+rac{ω_m(z)}{z^p} Q_{N-m}(z)$, with $P_m$ determined by interpolation and $Q_{N-m}$ by least-squares, and provides guidance for selecting the subspace dimension $r$ and the index $m$. Numerical experiments with the Rogers–Szegő weight show improvements over Szegő-type quadratures and pure interpolants, demonstrating the method’s potential when data are discrete and possibly nonuniform. The approach offers a robust, practical alternative for unit-circle integration in applications where samples are limited and traditional quadrature rules may be unstable or inapplicable.

Abstract

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.

Figures (3)

• Figure 1: The set of nodes $Z_{10}$, $\Xi_m$ and $\Upsilon_m$ in some particular cases.
• Figure 2: The set of nodes $Z_{15}$, $\Xi_m$ and $\Upsilon_m$ in some particular cases for $q \sim 1^{-}$.
• Figure 3: The optimal partition $\tilde{\Delta}_k$ for some particular choices of the parameters $m$,$q$ and $N$, along with the computed parameter $K_{\tilde{\Delta}_k}$ in (\ref{['Kopt']}) that measures the deviation of $\tilde{\Delta}_k$ from the uniform distribution.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Theorem 5
• Proposition 6
• Proposition 7