An Extension of a Result of Rivlin on Walsh Equiconvergence (Faber Nodes)

  • R. Brück
  • A. Sharma
  • R. S. Varga
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)

Abstract

We continue our investigations of generalizations of Walsh’s equiconvergence theorem. The setting is a compact set E of the complex plane, whose complement is simply connected in the extended complex plane, and the Faber polynomials associated with E. Here, we study equiconvergence phenomena for differences of interpolating polynomials, defined by Lagrange (and Hermite) interpolants in zeros of associated Faber polynomials.

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References

  1. [1]
    Brück R. On the failure of Walsh’s equiconvergence theorem for Jordan domains. Analysis, 13: 229–234, 1993.MathSciNetMATHGoogle Scholar
  2. [2]
    Brück R., Sharma A., Varga R. S. An extension of a result of Rivlin on Walsh equiconvergence. In Advances in Computational Mathematics: New Delhi, India, 1993, pages 225–234. World Scientific Publishing Co. Ptl. Ltd., Singapore, 1994. H. P. Dikshit and C. A. Micchelli, eds.Google Scholar
  3. [3]
    Cavaretta A. S. Jr., Sharma A., Varga R. S. Interpolation in the roots of unity: An extension of a theorem of J. L. Walsh. Resultate Math., 3: 155–191, 1980.MathSciNetMATHGoogle Scholar
  4. [4]
    Curtiss J. H. Convergence of complex Lagrange interpolation polynomials on the locus of the interpolation points. Duke Math. J., 32: 187–204, 1965.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Curtiss J. H. Faber polynomials and the Faber series. Amer. Math. Monthly, 78: 577–596, 1971.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Gaier D. Vorlesungen über Approximation im Komplexen. Birkhäuser Verlag, Basel, 1980.MATHGoogle Scholar
  7. [7]
    Kövari T., Pommerenke Ch. On Faber polynomials and Faber expansions. Math. Z., 99: 193–206, 1967.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Pommerenke Ch. Uber die Verteilung der Fekete-Punkte. Math. Ann., 168: 111–127, 1967.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Rivlin T. J. On Walsh equiconvergence. J. Approx. Theory, 36: 334–345, 1982.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Walsh J. L. Interpolation and Approximation by Rational Functions in the Complex Domain, 5th Ed. American Math. Soc., Providence, R.I., 1969.Google Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • R. Brück
    • 1
  • A. Sharma
    • 2
  • R. S. Varga
    • 3
  1. 1.Mathematisches InstitutJustus-Liebig-Universität GiessenGiessenGermany
  2. 2.Department of MathematicsUniversity of AlbertaEdmontonCanada
  3. 3.Institute for Computational MathematicsKent State UniversityKentUSA

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