Skip to main content
SpringerLink
Log in

Uniform rational approximation of |X|

  • Conference paper
  • First Online:
Methods of Approximation Theory in Complex Analysis and Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1550))

Abstract

We review recent results about best uniform rational approximants of |x| on [−1, 1]. We shall sketch the proof of the theorem that

where E nn (|x|,[−1, 1]) denotes the minimal approximation error. Related results are concerned with the asymptotic distribution of poles and zeros of the approximants r*n and the distribution of extreme points of the error function |x|−r*n(x) on [−1, 1]. Two conjectures about a generalisation of the approximation problem of |x| on [−1, 1] or of x α, α>0, on [0, 1] will be formulated.

Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299 14-1)

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 50.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.-E. Anderson, Rational approximation to functions like x α in integral norms, Anal. Math. 14 (1988), pp. 11–25.

    Article  MathSciNet  Google Scholar 

  2. S. Bernstein, Sur meilleure approximation de |x| par des polynômes de degrés donnés, Acta Math. 37 (1913), pp. 1–57.

    Article  MATH  Google Scholar 

  3. H.-P. Blatt, A. Iserles, and E.B. Saff, Remarks on the behavior of zeros and poles of best approximating polynomials and rational functions. In: Algorithms for Approximation (J.C. Mason and M.G. Cox, eds.), pp. 437–445. Inst. of Math. and Its Applic. Conference Series, Vol. 10, Claredon Press, Oxford, 1987.

    Google Scholar 

  4. A.P. Bulanow, The approximation of x 1/3 by rational functions, Vesci Akad. Navuk BSSR Ser. Fiz.-Navuk 1968, 2, pp. 47–56. (Russian)

    Google Scholar 

  5. A.P. Bulanow, Asymptotics for the least derivation of |x| from rational functions, Mat. Sb., 76 (118) (1968), pp. 288–303; English transl. in Math. USSR Sb. 5 (1968), pp. 275–290.

    MathSciNet  Google Scholar 

  6. G. Freud and J. Szabados, Rational Approximation to x α, Acta Math. Acad. Sci. Hungar. 18 (1967), pp. 393–393.

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Ganelius, Rational approximation to x α on [0, 1], Anal. Math. 5 (1979), pp. 19–33.

    Article  MathSciNet  MATH  Google Scholar 

  8. A.A. Gonchar, On the speed of rational approximation of continuous functions with characteristic singularities, Mat. Sb., 73 (115) (1967), pp. 630–638; English transl. in Math USSR Sb., 2 (1967).

    MathSciNet  MATH  Google Scholar 

  9. A.A. Gonchar, Rational approximation of the function x α, Constructive Theory of Functions (Proc. Internat. Conf., Varna 1970), Izdat. Bolgar. Akad. Nauk, Sofia, 1972, pp. 51–53. (Russian)

    Google Scholar 

  10. A.A. Gonchar, The rate of rational approximation and the property of single-valuedness of an analytic function in a neighborhood of an isolated singular point, Mat. Sb., 94 (136) (1974), pp. 265–282; English transl. in Math USSR Sb., 23 (1974).

    MathSciNet  Google Scholar 

  11. S. Järner, On weighted Zootarev problem for rational functions, Thesis, University of Götebrg, Göteborg, 1982

    Google Scholar 

  12. G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, New York 1967.

    Book  MATH  Google Scholar 

  13. D.J. Newman, Rational approximation to |x|, Mich. Math. J., 11 (1964), pp. 11–14.

    Article  MATH  Google Scholar 

  14. T.J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell Publ. Co., Waltham, Mass. 1969.

    MATH  Google Scholar 

  15. E.B. Saff, H. Stahl, and M. Wyneken, The Walsh table for x α, manuscript.

    Google Scholar 

  16. H.Stahl, Best uniform rational approximation of |x| on [−1, 1], to appear in Mat. Sb.

    Google Scholar 

  17. H.Stahl, Poles and zeros of best rational approximants of |x|, submitted to J. Comp.-Appl.Math.

    Google Scholar 

  18. J. Tzimbalario, Rational approximation to x α, J. App. Theory, 16 (1976), pp. 187–193.

    Article  MathSciNet  MATH  Google Scholar 

  19. R.S. Varga and A.J. Carpenter, On the Bernstein conjecture in approximation theory, Constr.Approx., 1 (1985), pp. 333–348; Russian transl. in Mat.Sb., 129 (171) (1986), pp. 535–548.

    Article  MathSciNet  MATH  Google Scholar 

  20. N.S. Vjacheslavov, The approximation of |x| by rational functions, Matem. Zametki 16 (1974), pp. 163–171, (Russian).

    MathSciNet  Google Scholar 

  21. N.S. Vjacheslavov, On the uniform approximation of |x| by rational functions, Dokl. Akad. Nauk SSSR, 220 (1975), pp. 512–515; English transl. in Soviet Math. Dokl., 16 (1975), pp. 100–104.

    MathSciNet  Google Scholar 

  22. N.S. Vjacheslavov, On the approximation of x α by rational functions, Izv. Akad. Nauk USSR, 44 (1980), English transl. in Math. USSR Izv. 16 (1981), pp. 83–101.

    Google Scholar 

  23. R.S. Varga, A. Ruttan and A.J. Carpenter, Numerical results on best uniform rational approximation of |x| on [−1, 1], to appear in Mat.Sb.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TFH-Berlin, Luxemburg str., 10/1000, Berlin 65, Germany

    Herbert Stahl

Authors
  1. Herbert Stahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Andrei A. Gonchar Edward B. Saff

Rights and permissions

Reprints and permissions

Copyright information

© 1993 The Euler International Mathematical Institute

About this paper

Cite this paper

Stahl, H. (1993). Uniform rational approximation of |X|. In: Gonchar, A.A., Saff, E.B. (eds) Methods of Approximation Theory in Complex Analysis and Mathematical Physics. Lecture Notes in Mathematics, vol 1550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0117477

Download citation

  • DOI: https://doi.org/10.1007/BFb0117477

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56931-2

  • Online ISBN: 978-3-540-47792-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation