Complex Analysis and Operator Theory

, Volume 5, Issue 3, pp 671–681 | Cite as

Cm-Approximation by Polyanalytic Polynomials on Compact Subsets of the Complex Plane

Article

Abstract

Several necessary and sufficient conditions on Cm-approximability of functions on compact subsets of the complex plane by polyanalytic polynomials are obtained. The resulting conditions for approximation are of the topological nature.

Keywords

Polyanalytic functions and polynomials Cm-approximation 

Mathematics Subject Classification (2000)

Primary 30E10 Secondary 30G30 

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References

  1. 1.
    Paramonov P.V.: Cm-approximations by harmonic polynomials on compact sets from \({\mathbb{R}^n}\). Russ. Acad. Sci. Sb. Math. 78(1), 231–251 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Paramonov P.V.: On approximation by harmonic polynomials in the C1-norm on compact sets in \({\mathbb{R}^2}\). Russ. Acad. Sci. Izv. Math. 42(2), 321–331 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Paramonov P.V., Fedorovskiy K.Yu.: Uniform and C1-approximability of functions on compact subsets of \({\mathbb{R}^2}\) by solutions of second order elliptic equations. Sb. Math. 190(2), 285–307 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Mergelyan S.N.: Uniform approximation to functions of a complex variables. Uspekhi Mat. Nauk (Russia Math. Surveys) 7(2), 31–122 (1952)MathSciNetGoogle Scholar
  5. 5.
    Carmona J.J.: Mergelyan approximation theorem for rational modules. J. Approx. Theory 44, 113–126 (1985)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fedorovski K.Yu.: Uniform n-analytic polynomial approximations of functions on rectifiable contours in \({\mathbb{C}}\) . Math. Notes 59(4), 435–439 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carmona J.J., Fedorovskiy K.Yu., Paramonov P.V.: On uniform approximation by polyanalytic polynomials and Dirichlet problem for bianalytic functions. Sb. Math. 193(10), 1469–1492 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Boivin A., Gauthier P.M., Paramonov P.V.: Uniform approximation on closed subsets of \({\mathbb{C}}\) by polyanalytic functions. Izv. Math. 68(3), 447–459 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carmona J.J., Fedorovskiy K.Yu.: Conformal maps and uniform approximation by polyanalytic functions. Oper. Theory Adv. Appl. 158, 109–130 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fedorovskiy K.Yu.: On some properties and examples of Nevanlinna domains. Proc. Steklov Inst. Math. 253, 186–194 (2006)CrossRefGoogle Scholar
  11. 11.
    Carmona J.J., Fedorovskiy K.Yu.: On the dependence of uniform polyanalytic polynomial approximation on the order of polyanalyticity. Math. Notes 83(1), 31–36 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Balk, M.B.: Polyanalytic functions. In: Mathematical Research, vol. 63. Academie Verlag, Berlin (1991)Google Scholar
  13. 13.
    Tarkhanov, N.N.: 1997 The analysis of solutions of elliptic equations. In: Mathematics and its Applications, vol. 406. Kluwer Academic Publishers, Dordrecht (1997)Google Scholar
  14. 14.
    Vitushkin, A.G.: Analytic capacity of sets in problems of approximation theory. Uspekhi Mat. Nauk, 22(6), 141–199 (1967) (Russian); in Russian Math. Surveys 22, 139–200 (1967)Google Scholar
  15. 15.
    Narasimhan, R.: Analysis on real and complex manifolds. In: Advanced Studies in Pure Mathematics, vol. 1. North-Holland, Amsterdam (1968)Google Scholar
  16. 16.
    Verdera J.: Cm-approximation by solution of elliptic equations and Calderon–Zygmund operators. Duke Math. J. 55(1), 157–187 (1987)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Moscow State Technical UniversityMoscowRussia

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