Numerische Mathematik

, Volume 57, Issue 1, pp 123–138 | Cite as

Characterization of the speed of convergence of the trapezoidal rule

  • Qazi I. Rahman
  • Gerhard Schmeisser
Article

Summary

Our aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.

Subject Classifications

AMS(MOS): 41A55, 65D30, 46E10 CR: G1.4 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bary, N.K.: A treatise on trigonometric series (Vol I and II). Oxford: Pergamon Press 1964Google Scholar
  2. 2.
    Blakeley, G.R., Borosh, I., Chui, C.K.: A two-dimensional mean problem. J. Approx. Theory22, 11–26 (1973)Google Scholar
  3. 3.
    Boas, R.P., Jr.: Entire functions. New York: Academic Press 1954Google Scholar
  4. 4.
    Brass, H.: Umkehrsätze beim Trapezverfahren. Aequationes Math.18, 338–344 (1978)Google Scholar
  5. 5.
    Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation (Vol. I). Basel, Birkhäuser 1971Google Scholar
  6. 6.
    Davis, P.J., Rabinowitz, P.: Methods of numerical integration (2nd edition). New York: Academic Press 1984Google Scholar
  7. 7.
    Loxton, J.H., Sanders, J.W.: On an inversion theorem of Möbius. J. Aust. Math. Soc., Ser. A30, 15–32 (1980)Google Scholar
  8. 8.
    Loxton, J.H., Sanders, J.W.: The kernel of a rule of approximate integration. J. Aust. Math. Soc., Ser. B21, 257–267 (1980)Google Scholar
  9. 9.
    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis (Vol. I and II, 4th edition), Berlin: Springer 1970 and 1971Google Scholar
  10. 10.
    Rahman, Q.I., Schmeisser, G.: Characterization of functions in terms of rate of convergence of a quadratur process. (Submitted to Proc. Amer. Math. Soc.)Google Scholar
  11. 11.
    Winter, A.: Diophantine approximations and Hilbert's space. Am. J. Math.66, 564–578 (1944)Google Scholar
  12. 12.
    Žensykbaev, A.A.: Best quadrature formula for some classes of periodic differentiable functions. Math. USSR Izy.11, 1055–1071 (1977)Google Scholar
  13. 13.
    Žensykbaev, A.A.: Best quadrature formula for the class\(W_{L_2 }^r \). Anal. Math.3, 83–95 (1977)Google Scholar
  14. 14.
    Zygmund, A.: Trigonometric series (Vol. I and II, 2nd edition). Cambridge: University Press 1968Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Qazi I. Rahman
    • 1
  • Gerhard Schmeisser
    • 2
  1. 1.Départment de mathématiques et de statistiqueUniversité de MontréalMontréalCanada
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenFederal Republic of Germany

Personalised recommendations