Approximation of the Hilbert Transform on the unit circle
This work provides an incremental improvement in numerical approximation of the Hilbert transform for researchers in signal processing and applied mathematics.
The paper presents a numerical method for approximating the Hilbert transform on the unit circle using Szegö and anti-Szegö quadrature formulas, achieving improved accuracy by averaging oppositely signed errors and strategically selecting a free parameter to distance nodes from the singularity. Numerical experiments demonstrate the method's accuracy.
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.