Two new formulas for the numerical evaluation of the Hilbert transform
type: Item
instance of: scholarly article
title: Two new formulas for the numerical evaluation of the Hilbert transform
zbMATH Open document ID: 0812.65132
DOI: 10.1007/BF01935644
author: Biancamaria Della Vecchia
published in: BIT
publication date: 1995-05-18T00:00:00Z
review text: The author develops two algorithms for the numerical evaluation of the Cauchy principal value integrals of the form: \[ H^{\alpha,\beta}[f](y)= \int^ \infty_ 0 {x^ \alpha\over (1+ x)^ \beta} {f(x)\over x- y} dx, \] with \(y\in (0,\infty)\) and \(f\) being Hölder continuous on \([0,R]\), \(R>0\). The first algorithm is related to a Gauss-Jacobi type quadrature rule recently developed by \textit{W. Gautschi} [BIT 31, No. 3, 438-446 (1991; Zbl 0734.65007)] for half-infinite intervals where \(f\in C^ 1[0,\infty)\) is required. The second one is based on a rational Bernstein-type operator introduced by \textit{G. Bleimann}, \textit{P. L. Butzer} and \textit{L. Hahn} [Indagationes Math. 42, 255-262 (1980; Zbl 0437.41021)]. Some numerical examples are given.
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 698692
zbMATH Keywords: semi-infinite Hilbert transforms
zbMATH Keywords: algorithms
zbMATH Keywords: Cauchy principal value integrals
zbMATH Keywords: Gauss-Jacobi type quadrature rule
zbMATH Keywords: rational Bernstein-type operator
zbMATH Keywords: numerical examples
reviewed by: Nácere Hayek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 698692