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Error Estimates for the Carathéodory-Fejér Method in Polynomial Approximation

  • Manfred Hollenhorst
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

In the Carathéodory-Fejér method one computes — starting from a complex power series absolutely convergent on the unit disk — a polynomial which is (hopefully) a better approximation to the function given by the power series than the truncated power series (with respect to the supremum norm). In this article we show that — under fairly restrictive conditions on the coefficients of the power series — the Carathéodory-Fejér method gives an asymptotically optimal approximation and in some cases is really a better approximation than the truncated power series.

Keywords

Power Series Polynomial Approximation Blaschke Product Supremum Norm Complex Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Carathéodory, C., and Fejér, L., Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und giber den PicardLandauschen Satz, Rendiconti del circolo matematico di Palermo, Vol. 32, (1911), 218–239.zbMATHCrossRefGoogle Scholar
  2. [2]
    Darlington, S., Analytical Approximations to Approximations in the Chebyshev Sense, The Bell System Technical Journal, VOL. 49, ( 1970, 1–32.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Gutknecht, M.H., Rational Carathéodory-Fejér approximation on a disk, a circle, and an interval, Journal of Approximation Theory, VOL. 41, (1984), 257–278.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Gutknecht, M.H. and Trefethen, L.N., Real polynomial Chebyshev approximation by the Carathéodory-Fejér method, SIAM Journal of Numerical Analysis, VOL. 19, (1982), 358–371.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Hollenhorst, M., Nichtlineare Verfahren bei der Polynomapproximation, Dissertation, University of Erlangen-Nürnberg, 1976.Google Scholar
  6. [6]
    Hurwitz, A., Über einen Satz des Herrn Kakeya,Tohoku Mathematical Journal, VOL. 4, (1913–1914), 89–93.Google Scholar
  7. [7]
    Jungen, R., Sur les séries de Taylor n’ayant que des singularités algébrologarithmiques sur leur cercle de convergence, Comm. Math. Helvet., VOL. 3, (1931), 266–306.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Klotz, V., Polynomiale und rationale Tschebyscheff-Approximation in der komplexen Ebene, Dissertation, University of Erlangen-Nürnberg, (1974).Google Scholar
  9. [9]
    Riordan, J., An Introduction to Combinatorial Analysis, New York, London, Sydney: John Wiley, (1958).Google Scholar
  10. [10]
    Saff, E.B. and Totik, V., Limitations of the Carathéodory-Fejér Method for Polynomial Approximation, Journal of Approximation Theory, VOL. 58, (1989), 284–296.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Specht, W. Enzyklopädie der mathematischen Wissenschaften. Band I, 1. Teil, Heft 8: Algebraische Gleichungen mit reellen oder komplexen Koeffizienten, Stuttgart: Teubner, (1958).Google Scholar
  12. [12]
    Varga, R.S., Matrix Iterative Analysis, Englewood Cliffs, N.J.: Prentice Hall, (1962).Google Scholar

Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • Manfred Hollenhorst
    • 1
  1. 1.Computer CenterJustus Liebig UniversityGiessenGermany

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