Skip to main content
SpringerLink
Log in

Rational approximation of Stieltjes functions by the Carathéodory-Fejér method

  • Published:
Constructive Approximation Aims and scope

Abstract

The Carathéodory-Fejér method provides a way for the construction of near best rational approximations. Gutknecht and Trefethen [7], [10] observed that in many cases the constructed functions yield very good estimates of the degree of rational approximation. We will Show that the correct asymptotic is obtained when the method is applied to Stieltjes functions. This fact is of interest in connection with Magnus' considerations of the 1/9-conjecture [9].

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. W. Barrett (1971):On the convergence of rational approximations to analytic functions of a certain class. J. Inst. Math. Applic.,7:308–323.

    Google Scholar 

  2. D. Braess (1986): Nonlinear Approximation Theory. Berlin, Heidelberg, New York: Springer-Verlag.

    Google Scholar 

  3. C. Carathéodory, L. Fejér (1911):Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz. Rend. Circ. Mat. Palermo,32:218–239.

    Google Scholar 

  4. T. Ganelius (1982):Degree of rational approximation. In: Lectures on Approximation and Value Distribution. Les Presses de l'Université de Montréal, No. 79.

  5. A. A. Gončar (1987):On the speed of rational approximation of some analytic functions. Mat. Sb.=Math. USSR-Sb.,34:131–145.

    Google Scholar 

  6. W. Gragg (1972):The Padé table and its relation to certain algorithms of numerical analysis. SIAM Rev.,14:1–61.

    Google Scholar 

  7. M. H. Gutknecht, L. N. Trefethen (1983):The Carathéodory-Fejér method for real rational approximation. SIAM J. Numer. Anal.,20:420–436.

    Google Scholar 

  8. H. L. Loeb, H. Werner (1974):Optimal numerical quadrature in H p -spaces. Math. Z.,138:111–117.

    Google Scholar 

  9. A. Magnus (submitted):Conjectured value of solution of the “1/9” problem by the CFGTmethod.

  10. L. N. Trefethen (1981):Near circularity of the error curve in complex Chebyshev approximation. J. Approx. Theory,31:344–367.

    Google Scholar 

  11. N. L. Trefethen (1981):Rational Chebyshev approximation on the unit disk. Numer. Math.,37:297–320.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität, 4630, Bochum, F.R.G.

    Dietrich Braess

Authors
  1. Dietrich Braess
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. Milne Anderson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Braess, D. Rational approximation of Stieltjes functions by the Carathéodory-Fejér method. Constr. Approx 3, 43–50 (1987). https://doi.org/10.1007/BF01890552

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01890552

Key words and phrases

AMS classification

Search

Navigation