Approximation of the Hilbert Transform on the unit circle

TL;DR

This paper introduces a numerical method for approximating the Hilbert transform on the unit circle using Szeg"o and anti-Szeg"o quadrature formulas, achieving high accuracy by strategic node placement and averaging techniques.

Contribution

The paper develops a novel approach employing Szeg"o and anti-Szeg"o quadrature formulas with optimized parameters to improve the approximation of the Hilbert transform.

Findings

Numerical experiments confirm high accuracy of the proposed method.

The method effectively reduces errors through averaging of quadrature formulas.

Nodes are chosen to avoid the singularity, enhancing precision.

Abstract

The paper deals with the numerical approximation of the Hilbert transform on the unit circle using Szeg\"o and anti-Szeg\"o quadrature formulas. These schemes exhibit maximum precision with oppositely signed errors and allow for improved accuracy through their averaged results. Their computation involves a free parameter associated with the corresponding para-orthogonal polynomials. Here, it is suitably chosen to construct a Szeg\"o and anti-Szeg\"o formula whose nodes are strategically distanced from the singularity of the Hilbert kernel. Numerical experiments demonstrate the accuracy of the proposed method.