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

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Approximation of the Hilbert Transform On The Unit Circle

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月10日

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  • Luisa Fermo,
  • Valerio Loi

摘要

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