Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals

Lubinsky, D. S.; Sidi, Avram

doi:10.1090/S0025-5718-1986-0815845-4

Convergence of product integration rules for functions with interior and endpoint singularities over bounded and unbounded intervals
HTML articles powered by AMS MathViewer

by D. S. Lubinsky and Avram Sidi PDF

Math. Comp. 46 (1986), 229-245 Request permission

Abstract:

The convergence of product integration rules, based on Gaussian quadrature points, is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals. The investigation is based on a new convergence result for Lagrangian interpolation and Gaussian quadrature of singular integrands.

References

Philip J. Davis and Philip Rabinowitz, Ignoring the singularity in approximate integration, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 367–383. MR 195256

Introduction to Measure Theory

Henning Esser, Bemerkungen zu einem Satz von G. Freud über Quadraturkonvergenz, Computing (Arch. Elektron. Rechnen) 8 (1971), 216–220 (German, with English summary). MR 300439, DOI 10.1007/bf02234104

Orthogonal Polynomials

Géza Freud, On Markov-Bernstein-type inequalities and their applications, J. Approximation Theory 19 (1977), no. 1, 22–37. MR 425426, DOI 10.1016/0021-9045(77)90026-0
D. S. Lubinsky, Gaussian quadrature, weights on the whole real line and even entire functions with nonnegative even order derivatives, J. Approx. Theory 46 (1986), no. 3, 297–313. MR 840397, DOI 10.1016/0021-9045(86)90067-5
D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Math. Comp. 43 (1984), no. 167, 219–242. MR 744932, DOI 10.1090/S0025-5718-1984-0744932-2
Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
Paul G. Nevai, Mean convergence of Lagrange interpolation. II, J. Approx. Theory 30 (1980), no. 4, 263–276. MR 616953, DOI 10.1016/0021-9045(80)90030-1
Philip Rabinowitz, Gaussian integration in the presence of a singularity, SIAM J. Numer. Anal. 4 (1967), 191–201. MR 213016, DOI 10.1137/0704018
Philip Rabinowitz and Ian H. Sloan, Product integration in the presence of a singularity, SIAM J. Numer. Anal. 21 (1984), no. 1, 149–166. MR 731219, DOI 10.1137/0721010
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
Ian H. Sloan and William E. Smith, Properties of interpolatory product integration rules, SIAM J. Numer. Anal. 19 (1982), no. 2, 427–442. MR 650061, DOI 10.1137/0719027
William E. Smith, Ian H. Sloan, and Alex H. Opie, Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials, Math. Comp. 40 (1983), no. 162, 519–535. MR 689468, DOI 10.1090/S0025-5718-1983-0689468-1