Skip to main content
SpringerLink
Log in

Error bound for product quadrature rules inL 1-weighted norm

  • Published:
CALCOLO Aims and scope Submit manuscript

Abstract

The authors study the error of the product quadrature rules to compute the integral\(\int_{ - 1}^1 {f\left( x \right) u\left( x \right)dx} \). Estimates inL 1-weighted norm are established whenu is a weight with algebraic and/or logarithmic singularities and the quadrature points are classical Jacobi zeros. Upper bounds for the generalized functions of second kind are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Braß,Quadraturverfahren, (1977), Vandenhoeck and Ruprecht, Göttingen.

    MATH  Google Scholar 

  2. G. Criscuolo, G. Mastroianni,A bound for generalized Jacobi functions of second kind, Calcolo XXIV, II (1987), 193–198.

    Article  MathSciNet  Google Scholar 

  3. G. Criscuolo, G. Mastroianni,Lagrange interpolation on generalized Jacobi with additional notes, Acta Mathematica Hungarica vol. 64 (1994), 27–49.

    Article  MathSciNet  Google Scholar 

  4. G. Criscuolo, G. Mastroianni, P. Nevai,Associated Generalized Jacobi Functions and Polynomials, J. Math. Anal. Appl. vol. 158, 1 (1991), 15–34.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. A. De Vore, R. Scott,Error bounds for Gaussian Quadrature and Weighted-L 1 Polynomial Approximation, SIAM J. Numer. Anal. 21, 2 (1984), 400–412.

    Article  MathSciNet  Google Scholar 

  6. Z. Ditzian, V. Totik,Moduli of smoothness, (1987), S. C. M., Springer-Verlag, Berlin.

    MATH  Google Scholar 

  7. K. J. Föster, K. Petras,On a Problem Proposed by H. Braß Concerning the Remainder Term in Quadrature for Convex Functions, Numer. Math. 57 (1990), 763–777.

    Article  MathSciNet  Google Scholar 

  8. G. Freud,Orthogonal Polynomials, (1971), Pergamon Press, New York.

    Google Scholar 

  9. G. Mastroianni,Generalized Christoffel Functions and Error of Positive Quadrature, to appear in Annals of Numer. Math.

  10. G. Mastroianni, P. Vertesi,Error estimates of product quadrature formulae, Numerical Integration IV ISNM vol. 112, Birkhäuser (1993), 241–252.

  11. G. Mastroianni, P. Vertesi,Weighted L p error of Lagrange interpolation, to appear in J. Approx. Theory.

  12. P. Nevai,Mean Convergence of Lagrange Interpolation III, Trans. Amer. Math. Soc. 282 (1984), 669–698.

    Article  MATH  MathSciNet  Google Scholar 

  13. K. Petras,Asymptotic Behaviour of Peano Kernel of Fixed Order, Numerical Integration III, (1988) (H. Braß, G. Hämmerlin eds.), ISNM 85, Birkäuser Verlag, Basel, 186–198.

    Google Scholar 

  14. K. Petras,Error Bounds for Gaussian and Related Quadrature and Application to R-Convex Functions, SIAM J. Numer. Anal. 29, 2 (1992), 578–585.

    Article  MATH  MathSciNet  Google Scholar 

  15. I. H. Sloan, W. E. Smith,Product-integration rules based on the zeros of Jacobi Polynomials, SIAM J. Numer. Anal. 17 (1980), 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Szegö,Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. vol. 23, Amer. Math. Soc., Providence, R. I. (1967).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Monte S. Angelo-Edificio T-Via Cintia, 80126, Napoli, Italy

    G. Criscuolo & L. Scuderi

Authors
  1. G. Criscuolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Scuderi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This material is based upon work supported by the «Ministero dell'Università e della Ricerca Scientifica e Tecnologica» (first author) and by the Italian Research Council (second author).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Criscuolo, G., Scuderi, L. Error bound for product quadrature rules inL 1-weighted norm. Calcolo 31, 73–93 (1994). https://doi.org/10.1007/BF02575720

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02575720

Keywords

Search

Navigation