Approximation of the Hilbert Transform on the unit circle

Abstract

The paper deals with the numerical approximation of the Hilbert transform on the unit circle using Szegö and anti-Szegö 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 Szegö and anti-Szegö formula whose nodes are strategically distanced from the singularity of the Hilbert kernel. Numerical experiments demonstrate the accuracy of the proposed method.