Advertisement

Numerische Mathematik

, Volume 58, Issue 1, pp 273–286 | Cite as

Gauss-Tchebycheff quadrature formulas

  • Franz Peherstorfer
Article

Summary

It is well known that the Tchebycheff weight function (1-x2)−1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln∈ℕ. In this paper we describe explicitly all weight functions which have the property that thenk-point Gauss quadrature formula has equal weights for allk∈ℕ, where (nk),n1<n2<..., is an arbitrary subsequence of ℕ. Furthermore results on the possibility of Tchebycheff quadrature on several intervals are given.

Subject classifications

AMS(MOS): 65D32 CR: G:1.4 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achieser, N.I.: Vorlesungen über Approximationstheorie. Berlin: Akademie-Verlag 1953Google Scholar
  2. 2.
    Brass, H.: Quadraturverfahren. Göttingen: Vanderhoeck und Ruprecht 1977Google Scholar
  3. 3.
    Förster, K.-J.: On Chebyshev quadrature for a special class of weight function. BIT26, 327–332 (1986)Google Scholar
  4. 4.
    Gautschi, W.: On some orthogonal polynomials of interest in theoretical chemistry. BIT24, 473–483 (1984)Google Scholar
  5. 5.
    Geronimo, J.S., Van Assche, W.: Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Amer. Math. Soc.308, 559–581 (1988)Google Scholar
  6. 6.
    Geronimus Ya.L.: On some distribution functions connected with systems of polynomials. C.R. (Doklady) Acad. Sci. URSS44, 355–359 (1944)Google Scholar
  7. 7.
    Geronimus, Ya.L.: On the Chebyshev quadrature formula. Math. USSR-Izv.3, 1115–1138 (1969)Google Scholar
  8. 8.
    Kahaner, D.K.: On equal and almost equal weight quadrature formulas. SIAM. J. Numer. Anal.6, 551–556 (1968)Google Scholar
  9. 9.
    Muschelischwili, N.I.: Singuläre Integralgleichungen Berlin: Akademie-Verlag 1965Google Scholar
  10. 10.
    Peherstorfer, F.: On Tchebycheff quadrature formulas. In: Numerical integration III, Braß H., Hämmerlin, G., (eds.) pp. 172–186 Basel: Birkhäuser 1988Google Scholar
  11. 11.
    Peherstorfer, F.: On Gauss quadrature formulas with equal weights. Numer. Math.52, 317–327 (1988)Google Scholar
  12. 12.
    Peherstorfer, F.: On Bernstein-Szegö orthogonal polynomials on several intervals II: Orthogonal polynomials with periodic recurrence coefficients. J. Approx. Theory (to appear)Google Scholar
  13. 13.
    Stahl, H., Totik, V.:N-th root asymptotic behaviour of orthonormal polynomials. In: Nevai, P. (ed.) Orthogonal polynomials, theory and practice, pp. 395–418. Dordrecht: Kluwer Academic 1989Google Scholar
  14. 14.
    Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Advances Math.3, 127–232 (1969)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Franz Peherstorfer
    • 1
  1. 1.Institut für MathematikJohannes Kepler Universität LinzLinzAustria

Personalised recommendations