Advertisement

Abstract

Two Contributions to Numerical Quadrature. Let G be a domain in the complex plane containing the unit interval I. In the first contribution we consider bounds for the remainder R(f) of a quadrature formula on I which are of the form |R(f)|≤ c-sup {|f(z)}: zϵG}. We ask for the best possible constant c, study-its structure, and give an approach to calculate its asymptotically precise value for large domain G or large number of nodes. For quadrature formulae of order 2 explicit results are obtained. Furthermore, while looking on a problem raised by Brass we find a new global bound for c in the case of some formulae of order 2. In the second contribution we discuss a question asked by Collatz and concerning the choice of an error estimate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albrecht, J. Private communication, October 1978.Google Scholar
  2. 2.
    Brass, H.: Quadraturverfahren. Göttingen-Zürich, Vandenhoeck & Ruprecht 1977.Google Scholar
  3. 3.
    Davis, P.J.: “Errors of numerical approximation for analytic functions”. In: J. Todd: A survey of numerical analysis, 468–484. New York, Mc Graw-Hill 1962.Google Scholar
  4. 4.
    Düren, P.L.: Theory of HP spaces. New York — London, Academic Press 1970.Google Scholar
  5. 5.
    Hämmerlin, G.: Über ableitungsfreie Schranken für Quadraturfehler. Numer. Math. 5(1963), 226–233 and 7(1965), 232–237.CrossRefGoogle Scholar
  6. 6.
    Krzyz, J.: Problems in complex variable theory. New York, American Elsevier Publishing Company, Inc. 1971.Google Scholar
  7. 7.
    Labelle, G.: Concerning polynomials on the unit interval. Proc. Amer. Math. Soc. 20 (1969), 321–326.CrossRefGoogle Scholar
  8. 8.
    Oberhettinger, F. und W. Magnus: Anwendung der elliptischen Funktionen in Physik und Technik. Berlin-Göttingen-Heidelberg, Springer 1949.CrossRefGoogle Scholar
  9. 9.
    Rahman, Q.I.: Some inequalities concerning functions of exponential type. Trans. Amer. Math. Soc. 135 (1969), 281–293.CrossRefGoogle Scholar
  10. 10.
    Riesz, F.: Über Potenzreihen mit vorgeschriebenen Anfangsgliedern. Acta Math. 42 (1920), 145–171.CrossRefGoogle Scholar
  11. 11.
    Szász, O.: Ungleichheitsbeziehungen für die Ableitung einer Potenzreihe, die eine im Einheitskreise beschränkte Funktion darstellt. Math. Z. 8 (1920), 303–309.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Gerhard Schmeisser
    • 1
  1. 1.Mathematisches InstitutErlangenGermany

Personalised recommendations