Special Topics

  • Vladimir V. Andrievskii
  • Hans-Peter Blatt
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we consider the approximation of a piecewise analytic function on touching domains by polynomials and the convergence of Bieberbach polynomials in domains with quasiconformal boundary. We have already used some of the results of this chapter in previous sections.

Keywords

Special Topic Polynomial Approximation Singular Integral Operator Polynomial Kernel Admissible Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Historical Comments

  1. [16]
    V.V. Andrievskii (1996): Polynomial approximation to a piecewise analytic function on touching domains. Complex Variables, 31: 325–344.MathSciNetCrossRefGoogle Scholar
  2. [80]
    R. Grothmann, E.B. Saff (1988): On the behavior of zeros and poles of best uniform polynomial and rational approximants. Nonlinear numerical methods and rational approximation (ed. A. Cuyt), Reidel, Dordrecht: 57–75.Google Scholar
  3. [8]
    J.M. Anderson, W.H.J. Fuchs (1988): Remarks on “almost best” approximation in the complex plane. Complex Analysis (ed. J. Hersch, A. Huber ): 17–25.Google Scholar
  4. [154]
    E.B. Saff, V. Totik (1989): Polynomial approximation of piecewise analytic functions. J. London Math. Soc., 39: 487–498.MathSciNetCrossRefMATHGoogle Scholar
  5. [66]
    D. Gaier (1997): Complex approximation on touching domains. Complex Variables, 34: 325–342.MathSciNetCrossRefMATHGoogle Scholar
  6. [93]
    M.V. Keldysh (1939): Sur l’approximation en moyenne quadratique des fonctions analytiques. Mat. Sb., 5: 391–401.MATHGoogle Scholar
  7. [123]
    S.N. Mergelyan (1951): Some questions in constructive function theory. Tr. Mat. Inst. Steklova, Akad. Nauk SSSR, 37: 3–91 (Russian).Google Scholar
  8. [179]
    Wu Xue-Mou (1963): On Bieberbach polynomials. Acta Math. Sinica, 13: 145–151.MATHGoogle Scholar
  9. [167]
    P.K. Suetin (1971): Polynomials orthogonal over a region and Bieberbach polynomials. Proc. Steklov Inst. Math., 100. Providence, RI 1974. American Math. Society.Google Scholar
  10. [162]
    I.B. Simonenko (1978): Convergence of Bieberbach polynomials in the case of a Lipschitz domain. Math USSR-Izv., 13: 166–174.CrossRefGoogle Scholar
  11. [11]
    V.V. Andrievskii (1984): Convergence of Bieberbach polynomials in domains with quasiconformal boundary. Ukrainian Math J., 32: 435–440 (Russian).Google Scholar
  12. [63]
    D. Gaier (1987): On a polynomial lemma of Andrievskii. Arch. Math., 49: 119–123.MathSciNetCrossRefMATHGoogle Scholar
  13. [64]
    D. Gaier (1988): On the convergence of Bieberbach polynomials in regions with corners. Constr. Approx., 4: 289–305.MathSciNetCrossRefMATHGoogle Scholar
  14. [65]
    D. Gaier (1992): On the convergence of Bieberbach polynomials in regions with piecewise analytic boundary. Arch. Math., 58: 462–470.MathSciNetCrossRefMATHGoogle Scholar
  15. [22]
    V.V. Andrievskii, D. Gaier (1992): Uniform convergence of Bieberbach polynomials in domains with piecewise quasianalytic boundary. Mitt. Math. Sem. Giessen, 211: 49–60.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Vladimir V. Andrievskii
    • 1
  • Hans-Peter Blatt
    • 2
  1. 1.Department of Mathematics and Computer ScienceKent State UniversityKentUSA
  2. 2.Mathematisch-Geographische FakultätKatholische Universität EichstättEichstättGermany

Personalised recommendations