Geodesic
On~approximation by harmonic polynomials in the $C^1$-norm on compact sets in~$\mathbf R^2$
Izvestiya. Mathematics , Tome 42 (1994) no. 2, pp. 321-331.

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It is proved that for an arbitrary compact set $X$ in $\mathbf R^2$ the following conditions are equivalent: 1) for every function $f\in C^1(\mathbf R^2)$, harmonic on $X^0$, and for any $\varepsilon>0$ a harmonic polynomial $p$ can be found such that $$ \|f-p\|_X\varepsilon,\qquad \|\nabla(f-p)\|_X\varepsilon; $$ 2) the set $\mathbf R^2\setminus X$ is connected 
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P. V. Paramonov. On~approximation by harmonic polynomials in the $C^1$-norm on compact sets in~$\mathbf R^2$. Izvestiya. Mathematics , Tome 42 (1994) no. 2, pp. 321-331. http://geodesic.mathdoc.fr/item/IM2_1994_42_2_a3/

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