Some numerical algorithms to evaluate Hadamard finite-part integrals

Abstract Some algorithms are described for the numerical evaluation of Hadamard finite part integrals of type ǂ 1 −1 [f(x)/(x−t) 2 ]v α,β dx,|t| , where vα,β is a Jacobi weight. Convergence results and some numerical examples are given.

[1]  Giovanni Monegato,et al.  Numerical evaluation of hypersingular integrals , 1994 .

[2]  I. Gopengauz.  A theorem of A. F. Timan on the approximation of functions by polynomials on a finite segment , 1967 .

[3]  G. Mastroianni.  Uniform convergence of derivatives of Lagrange interpolation , 1992 .

[4]  David Elliott,et al.  An algorithm for the numerical evaluation of certain Cauchy principal value integrals , 1972 .

[5]  P. Nevai,et al.  Mean convergence of derivatives of Lagrange interpolation , 1991 .

[6]  V. Totik,et al.  Moduli of smoothness , 1987 .

[7]  G. P. Névai,et al.  Mean convergence of Lagrange interpolation, II☆ , 1976 .

[8]  D. F. Paget,et al.  The numerical evaluation of Hadamard finite-part integrals , 1981 .

[9]  G. Mastroianni,et al.  Interlacing properties of the zeros of the orthogonal polynomials and approximation of the Hilbert transform , 1995 .

[10]  Giuseppe Mastroianni,et al.  On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals , 1989 .

[11]  Giuseppe Mastroianni,et al.  Convergence of extended Lagrange interpolation , 1990 .

[12]  Giuseppe Mastroianni,et al.  Uniform convergence of derivatives of extended Lagrange interpolation , 1991 .

[13]  George J. Tsamasphyros,et al.  Gauss quadrature rules for finite part integrals , 1990 .

[14]  Walter Gautschi,et al.  Computing the Hilbert transform of a Jacobi weight function , 1987 .

[15]  G. Criscuolo,et al.  Convergenza di formule gaussiane per il calcolo delle derivate di integrali A valor principale secondo Cauchy , 1987 .

[16]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .