De Bonis, M. C.; Mastroianni, G. Some simple quadrature rules for evaluating the Hilbert transform on the real line. (English) Zbl 1056.65019 Arch. Inequal. Appl. 1, No. 3-4, 475-494 (2003). This paper deals with the numerical integration of special Cauchy principle value integrals on the real line. As basis for Gaussian quadrature formulas generalized Hermite or Markov-Sonin polynomials are used. A strategy for determining the degree of the orthogonal polynomial (equal to the total number of nodes) and for selecting the relevant and necessary nodes (no coincidence with the singularity point and a decrease of the number of nodes) is developed. An error analysis is given and error estimates are derived. Some numerical tests are shown. Reviewer: Rudolf Scherer (Karlsruhe) Cited in 5 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 44A15 Special integral transforms (Legendre, Hilbert, etc.) 65R10 Numerical methods for integral transforms Keywords:Hilbert transform; Cauchy principle value integrals; orthogonal polynomials; Gaussian quadrature formulas; error estimates; numerical examples; Markov-Sonin polynomials PDFBibTeX XMLCite \textit{M. C. De Bonis} and \textit{G. Mastroianni}, Arch. Inequal. Appl. 1, No. 3--4, 475--494 (2003; Zbl 1056.65019)