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Approximation Theory and Its Applications

, Volume 17, Issue 1, pp 97–105 | Cite as

On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

  • Abdullah Cavus
  • Fahreddin G. Abdullayev
Article
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Abstract

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z0be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r0>0 and centered at the origin 0, normalized by ϕ(z0) = 0 and ϕ(z0) = 1. Let us set\(\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta \), and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z0) that minimizes the integral\(\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }\)in the class\(\mathop \prod \limits_n \)of all polynomials of degreenand satisfying the conditions P n (z0) = 0 and P n (z0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ϕ p (z) on\(\bar G\)in case of\(p > 2 - \frac{{K^2 + 1}}{{2K^4 }}\).

Keywords

Complex Plane Conformal Mapping Uniform Convergence Formal Boundary Finite Domain 
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References

  1. [1]
    Abdullaryev, F. G., On the Orthogonal Polynomials in Domains with Quasiconformal Boundary (Russian). Dissertation, Donetsk (1986).Google Scholar
  2. [2]
    Abdullayev, F. G., On the Convergence of Bieberbach Polynomials In domains with Interior Zero Angles (Russian). Dokl. Akad. Nauk. Ukrain. SSR, Ser. A., 12(1989), 3–5.Google Scholar
  3. [3]
    Ahlfors, L. V., Lectures on Quasiconformal Mappings. Princeton, NJ: Van Nostrand(1966).Google Scholar
  4. [4]
    Andrievskii, V. V., Uniform Convergence of Bieberbach Polynomials in Domains with Zero Angles (Russian). Dokl. Akad. Nauk. Ukrain. SSR, Ser. An. 4(1982), 3–5.Google Scholar
  5. [5]
    Andrievskii, V. V., Convergence of Bieberbach Polynomials in Domains with Quasiconformal Boundary. Translation. Ukrainian Math. J., 32(1981), 295–299.Google Scholar
  6. [6]
    ANdrievskii, V. V., Uniform Convergence of Bieberbach Polynomials in Domains with Piecewise Quasiconformal Boundary (Russian). In: Theory of Mappings and Approximation of Functions. Kiev, Naukova Dumka(1983), 3–18.Google Scholar
  7. [7]
    Andrievskii, V. V., Constructive Characterization of the Harmonic Functions in Domains with Quasiconformal Boundary (Russian). Preprint, In: Quasiconformal Continuation and Approximation by Functions in the Set of the Complex Plane, Kiev(1985).Google Scholar
  8. [8]
    Batchaev, I. M., Integral Representations in a Region with a Quasiconformal Boundary and Some Applications (Russian). Dissertation, Baku(1981).Google Scholar
  9. [9]
    Belyi, V. I., Conformal Mappings and the Approximation of Analytic Functions in Domains with a Quasiconformal Boundary. math. USSR—sb., 31(1977), 289–317.Google Scholar
  10. [10]
    Bes, L., A Non-Standard Integral Equation with Applications to Quasiconformal Mappings. acta Math., 116(1966), 113–134.Google Scholar
  11. [11]
    Davis, P. J., Interpolation and Approximation. Blaisdell Publishing Company(1963).Google Scholar
  12. [12]
    Gaier, D., On the Convergence of the Bieberbach Polynomials in Regions with Corners. Constructive Approximation, 4(1988), 289–305.Google Scholar
  13. [13]
    Gaier, D., On the Convergence of the Bieberbach Polynomials in Regions with Piecewise-Analytic Boundary. arch. Math., 58(1992), 462–470.Google Scholar
  14. [14]
    Gehring, F. W. and Martio, O., Lipschitz Classes and Quasiconformal Mappings. Annal. acad. Scien. Fenn. Series A. 1. Mathematica, 10(1985), 203–219.Google Scholar
  15. [15]
    Goldstein, V. M., The Degree of Summability of Generalized Derivatives of Plane Quasiconformal Homeomorphisms. Soviet Math. Dokl., 21(1980), No. 1, 10–13.Google Scholar
  16. [16]
    Hinkanen, A. Anderson, J. M. and Gehring F. W., Polynomial Approximation in Quasi Discs. In: Differential Geometry and Complex Analysis, Edited by I. Chavely & H. M. Farkas, Springer-Verlag(1985), Berlin.Google Scholar
  17. [17]
    Israfilov, D. M., On the Approximation Properties of Extremal Polynomials (Russian). Dep. VINITI, 546(1981).Google Scholar
  18. [18]
    Keldych, M. V., Sur l' Approximation en Moyenne Quadratique des Fonctions Analytiques. Math. Sb. 47(1939), No. 5, 391–401.Google Scholar
  19. [19]
    Kulikov, I. V., L p-Convergence of Bieberbach Polynomials. Math. USSR-Izv., 15(1980), 349–371.Google Scholar
  20. [20]
    Kulikov, I. V., W 21-L Convergence of Bieberbach Polynomials in a Lipschitz Domain. Russian Math. Surveys, 36(1981), 161–162.Google Scholar
  21. [21]
    Leclerc, M., A Note on a Theorem of V. V. Andrievskii, Arch. Math. 46(1986), 159–161.Google Scholar
  22. [22]
    Lehto, O. and Virtanen, K. I., Quasiconformal Mappings in the Plane. Springer-Verlag(1973), Berlin.Google Scholar
  23. [23]
    Mergelyan, S. N., Certain Questions of the Constructive Theory of Functions (Russian), Trudy Math. Inst. Steklov, 37(1951).Google Scholar
  24. [24]
    Privalov, I. I., Introduction to the Theory of Functions of a Complex Variable. Moscow, Nauka (1984).Google Scholar
  25. [25]
    Simonenko, I. B., On the Convergence of Bieberbach Ploynomials in the case of a Lipschitz Domain. Math. USSR-Izv., 13(1980), 166–174.Google Scholar
  26. [26]
    Smirnov, V. I. and Lebedev, N. A., Functions of a Complex Variable. CONSTRUCTIVE THEORY. The M. I. T. PRESS(1968).Google Scholar
  27. [27]
    Suetin, P. K., Polynomials Orthogonal Over a Region and Bieberbach Polynomials. Proc. Steklov Inst. Math.100(1971). Providence, Rhode Island: Amer. Math. Soc., (1974).Google Scholar
  28. [28]
    Walsh, J. L., Interpolation and Approximation by Rational Functions in the Complex Domain (Russian). Moscow(1961).Google Scholar
  29. [29]
    Wu Xuemou, On Bieberbach Polynomials. Acta Math. Sinicia, 13(1963), 145–151.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Abdullah Cavus
    • 1
  • Fahreddin G. Abdullayev
    • 2
  1. 1.Faculty of Arts & Sciences, Department of MathematicsKaradeniz Technical UniversityTrabzonTurkey
  2. 2.Institute of Mathematics & MechanicsAcademy of Sciences of Azerbaijan RepublicBakuAzerbaijan

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