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. 
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/
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