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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     author = {A. B. Zaitsev},
     title = {On the {Closure} of the {Sum} of {Two} {Uniform} {Algebras} on {Compact} {Sets} in~$\mathbb C$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {34--42},
     publisher = {mathdoc},
     volume = {89},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a3/}
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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/