Mathematical Notes

, Volume 83, Issue 1–2, pp 31–36 | Cite as

On the dependence of uniform polyanalytic polynomial approximations on the order of polyanalyticity

Article

Abstract

In this paper, we construct, for each n ∈ ℕ, a compact set X ⊂ ℂ (depending on n) such that the set of all polyanalytic polynomials of order n is not dense in C(X), but the set of all polyanalytic polynomials of order 2n is already dense in C(X).

Key words

polyanalytic function polyanalytic polynomial uniform approximation Schwartz function Vandermonde matrix 

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References

  1. 1.
    J. J. Carmona, P. V. Paramonov, and K. Yu. Fedorovskii, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions,” Mat. Sb. 193(10), 75–98 (2002) [Russian Acad. Sci. Sb. Math. 193 (10), 1469–1492 (2002)].MathSciNetGoogle Scholar
  2. 2.
    K. Yu. Fedorovskii, “Uniform n-analytic polynomial approximations of functions on rectifiable contours in ℂ,” Mat. Zametki 59(4), 604–610 (1996) [Math. Notes 59 (3–4), 435–439 (1996)].MathSciNetGoogle Scholar
  3. 3.
    K. Yu. Fedorovskii, “On some properties and examples of Nevanlinna domains,” in Trudy Mat. Inst. Steklov, Vol. 253: Complex Analysis and Applications (Collection of Papers) (Nauka, Moscow, 2006), pp. 204–213 [Proc. Steklov Inst.Math., Vol. 253, pp. 186–194 (2006)].Google Scholar
  4. 4.
    A. Boivin, P. M. Gauthier, and P. V. Paramonov, “On uniform approximation by n-analytic functions on closed subsets in‒,” Izv. Ross. Akad. Nauk Ser. Mat. 68(3), 15–28 (2004) [Russian Acad. Sci. Izv. Math. 68 (3), 447–459 (2004)].MathSciNetGoogle Scholar
  5. 5.
    J. J. Carmona and K. Yu. Fedorovskiy, “Conformal maps and uniform approximation by polyanalytic functions,” in Selected Topics in Complex Analysis, Oper. Theory Adv. Appl. (Birkhäuser Verlag, Basel, 2005), Vol. 158, pp. 109–130.CrossRefGoogle Scholar
  6. 6.
    P. Davis, The Schwarz Function and Its Applications, in The Carus Math.Monographs (The Mathematical Association of America, Washington, DC, 1974), Vol. 17.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Universitat Autonoma de BarcelonaBarcelonaSpain
  2. 2.State University of ManagementMoscowRussia

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