Approximation of the Hilbert transform on $(0,+\infty)$ by using discrete de la Vall\'ee Poussin filtered polynomials

Year 2024, , 114 - 128, 16.12.2024

Donatella Occorsio

https://doi.org/10.33205/cma.1541668

Abstract

In the present paper, is proposed a method to approximate the Hilbert transform of a given function $f$ on $(0,\infty)$ employing truncated de la Vallée discrete polynomials recently studied in [25]. The method generalizes and improves in some sense a method based on truncated Lagrange interpolating polynomials introduced in [24], since is faster convergent and simpler to apply. Moreover, the additional parameter defining de la Vallée polynomials helps to attain better pointwise approximations. Stability and convergence are studied in weighted uniform spaces and some numerical tests are provided to asses the performance of the procedure.

Keywords

Hilbert transform, discrete de la Vallée-Poussin approximation, Approximation by polynomials

Supporting Institution

This work was partially supported by PRIN 2022 PNRR project no. P20229RMLB financed by the European Union - NextGeneration EU and by the Italian Ministry of University and Research (MUR).

Thanks

The research has been accomplished within RITA (Research ITalian network on Approximation), UMI-T.A.A. (Unione Matematica Italiana- Teoria dell’Approssimazione e Applicazioni), and ANA&A (Approssimazione Numerica ed Analitica di dati e di Funzioni con Applicazioni) working group.

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