Two new formulas for the numerical evaluation of the Hilbert transform
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.
scientific article; zbMATH DE number 698692
scientific article