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Approximation of the Hilbert Transform on the unit circle

Luisa Fermo, Valerio Loi

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.

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.
Paper Structure (10 sections, 7 theorems, 110 equations, 1 figure, 7 tables)

This paper contains 10 sections, 7 theorems, 110 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Quadrature schemes
  4. Szegö rules
  5. Anti-Szegö quadrature rules
  6. Averaged scheme
  7. Pointwise approximation of $Hf$
  8. An algorithm based on a prescribed node
  9. Numerical tests
  10. Conclusions

Key Result

Theorem 3.1

The $n$ zeros of the polynomial $\hat{\psi}_n(z;\tau)$ strictly interlace the zeros of the polynomial $\hat{\psi}_n(z;-\tau)$.

Figures (1)

  • Figure 1: Graph of the errors for Example \ref{['test1']} with $n=4$.

Theorems & Definitions (20)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 4.1
  • Theorem 4.2
  • proof
  • ...and 10 more