Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the remainder of Gaussian quadrature formulas for Bernstein-Szegő weight functions


Author: F. Peherstorfer
Journal: Math. Comp. 60 (1993), 317-325
MSC: Primary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1993-1153169-2
MathSciNet review: 1153169
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an explicit expression for the kernel of the error functional for Gaussian quadrature formulas with respect to weight functions of Bernstein-Szegö type, i.e., weight functions of the form $ {(1 - x)^\alpha }{(1 + x)^\beta }/\rho (x),\quad x \in ( - 1,1)$, where $ \alpha ,\beta \in \{ - \tfrac{1}{2},\tfrac{1}{2}\} $ and $ \rho $ is a polynomial of arbitrary degree which is positive on $ [ - 1,1]$. With the help of this result the norm of the error functional can easily be calculated explicitly for a wide subclass of these weight functions.


References [Enhancements On Off] (What's this?)

  • [1] G. Akrivis, The error norm of certain Gaussian quadrature formulae, Math. Comp. 45 (1985), 513-519. MR 804940 (87a:65051)
  • [2] W. Gautschi, On Padé approximants associated with Hamburger series, Calcolo 20 (1983), 111-127. MR 746349 (85f:41012)
  • [3] W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170-1186. MR 723834 (85j:65010)
  • [4] G. Hämmerlin, Fehlerabschätzungen bei numerischer Integration nach Gauss, Methoden und Verfahren der Mathematischen Physik, vol. 6 (B. Brosowski and E. Martensen, eds.), Bibliographisches Institut, Mannheim, Wien, Zürich, 1972, pp. 153-163. MR 0359277 (50:11732)
  • [5] R. Kumar, A class of quadrature formulas, Math. Comp. 28 (1974), 769-778. MR 0373240 (51:9441)
  • [6] -, Certain Gaussian quadratures, J. Inst. Math. Appl. 14 (1974), 175-182. MR 0356452 (50:8922)
  • [7] N. I. Muschelischwili, Singuläre Integralgleichungen, Akademie Verlag, Berlin, 1965. MR 0193453 (33:1673)
  • [8] S. E. Notaris, The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type, Numer. Math. 57 (1990), 271-283. MR 1057125 (92a:65080)
  • [9] F. Peherstorfer, On Bernstein-Szegö orthogonal polynomials on several intervals, SIAM J. Math. Anal. 21 (1990), 461-482. MR 1038902 (91b:42043)
  • [10] G. Szegö, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1967.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D32

Retrieve articles in all journals with MSC: 65D32


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1153169-2
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society