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On the closure of the sum of two uniform algebras on compact sets in ℂ

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Abstract

Necessary and sufficient conditions on a compact set X in ℂ and a self-homeomorphism ψ of the plane ℂ are studied under which any function continuous on X can be approximated uniformly on X by functions of the form p + h ∘ ψ, where p is a polynomial in a complex variable and h is a rational function whose poles belong to the bounded components of the complement to the compact set ψ(X).

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References

  1. M. A. Lavrent’ev, “On functions of a complex variable representable by series of polynomials,” in Selected Works: Mathematics and Mechanics (Nauka, Moscow, 1990), pp. 149–185 [in Russian].

    Google Scholar 

  2. A. G. Vitushkin, “Analytic Capacity of Sets in Problems of Approximation Theory,” Uspekhi Mat. Nauk 22(6), 141–199 (1967).

    MATH  Google Scholar 

  3. J. L. Walsh, “The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions,” Bull. Amer.Math. Soc. 35(4), 499–544 (1929).

    Article  MathSciNet  MATH  Google Scholar 

  4. T.W. Gamelin, Uniform Algebras (Prentice-Hall, Englewood Cliffs, NJ, 1969; Mir,Moscow, 1973).

    MATH  Google Scholar 

  5. A. G. O’Farrell, “A generalizedWalsh-Lebesgue theorem,” Proc. Roy. Soc. Edinburgh Sec. A 73(1), 231–234 (1975).

    MathSciNet  Google Scholar 

  6. A. B. Zaitsev, “On the uniform approximability of functions by polynomials of special classes on compact sets in R2,” Mat. Zametki 71(1), 75–87 (2002) [Math. Notes 71 (1), 68–79 (2002)].

    MathSciNet  Google Scholar 

  7. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972; Mir, Moscow, 1977).

    Google Scholar 

  8. E. Bishop, “Boundary measures of analytic differentials,” in Some Questions of Approximation Theory (Inostr. Lit., Moscow, 1963), pp. 87–100 [Duke.Math. J. 27 (3), 331–340 (1960).]

    Google Scholar 

  9. J. J. Carmona and K. Yu. Fedorovskiy, “Conformal maps and uniform approximation by polyanalytic functions,” in Oper. Theory. Adv. Appl., Vol. 158: Selected Topics in Complex Analysis (Birkh äuser, Basel, 2005), pp. 109–130.

    Google Scholar 

  10. A. Roth, “Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen,” Comment.Math. Helv. 11(1), 77–125 (1938).

    Article  MathSciNet  Google Scholar 

  11. S. N. Mergelyan, “Uniform Approximations of Functions of a Complex Variable,” UspekhiMat. Nauk 7(2), 31–122 (1952).

    MathSciNet  Google Scholar 

  12. I. I. Privalov, Boundary Properties of Analytic Functions (Gosudarstv. Izd. Tekhn.-Teor. Lit., Moscow-Leningrad, 1950) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Moscow State Institute of Radioengineering, Electronics, and Automation, Moscow, Russia

    A. B. Zaitsev

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  1. A. B. Zaitsev
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Correspondence to A. B. Zaitsev.

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Original Russian Text © A. B. Zaitsev, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 1, pp. 34–42.

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Zaitsev, A.B. On the closure of the sum of two uniform algebras on compact sets in ℂ. Math Notes 89, 51–58 (2011). https://doi.org/10.1134/S0001434611010056

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