Geodesic
On the Closure of the Sum of Two Uniform Algebras on Compact Sets in~$\mathbb C$
Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 34-42.

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Necessary and sufficient conditions on a compact set $X$ in $\mathbb C$ and a self-homeomorphism $\psi$ of the plane $\mathbb C$ are studied under which any function continuous on $X$ can be approximated uniformly on $X$ by functions of the form $p+h\circ\psi$, 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 $\psi(X)$.
Keywords: approximation of homeomorphisms of the complex plane, approximation by sums of polynomials and rational functions, uniform approximation, compact set without interior points with disconnected complement, harmonic measure. 
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A. B. Zaitsev. On the Closure of the Sum of Two Uniform Algebras on Compact Sets in~$\mathbb C$. Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 34-42. http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a3/

[1] M. A. Lavrentev, “O funktsiyakh kompleksnogo peremennogo, predstavimykh ryadami polinomov”, Izbrannye trudy. Matematika i mekhanika, Nauka, M., 1990, 149–185 | MR | Zbl

[2] A. G. Vitushkin, “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199 | MR | Zbl

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

[4] T. Gamelin, Ravnomernye algebry, Mir, M., 1973 | MR | Zbl

[5] A. G. O'Farrell, “A generalized Walsh–Lebesgue theorem”, Proc. Roy. Soc. Edinburgh Sect. A, 73:1 (1975), 231–234 | MR | Zbl

[6] A. B. Zaitsev, “O ravnomernoi priblizhaemosti funktsii polinomami spetsialnykh klassov na kompaktakh v $\mathbb R^2$”, Matem. zametki, 71:1 (2002), 75–87 | MR | Zbl

[7] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 1, Funktsionalnyi analiz, Mir, M., 1977 | MR | Zbl

[8] E. Bishop, “Granichnye mery analiticheskikh differentsialov”, Nekotorye voprosy teorii priblizhenii, IL, M., 1963, 87–100; E. Bishop, “Boundary measures of analytic differentials”, Duke. Math. J., 27:3 (1960), 331–340 | DOI | MR | Zbl

[9] J. J. Carmona, K. Yu. Fedorovskiy, “Conformal maps and uniform approximation by polyanalytic functions”, Selected Topics in Complex Analysis, Oper. Theory. Adv. Appl., 158, Birkhäuser, Basel, 2005, 109–130 | DOI | MR | Zbl

[10] A. Roth, “Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen”, Comment. Math. Helv., 11:1 (1938), 77–125 | DOI | MR | Zbl

[11] S. N. Mergelyan, “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 | MR | Zbl

[12] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, M.–L., 1950 | MR | Zbl