Approximation of the Hilbert Transform On The Unit Circle

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