Geodesic
Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in $\mathbb{R}^2$
Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 41-51.

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In this paper, we study necessary and sufficient conditions for functions to be approximated uniformly on plane compact sets by polynomial solutions to second-order homogeneous elliptic equations with constant coefficients. Sufficient conditions for approximability are of reductive character, i.e., the possibility of approximating on some (simpler) parts of the compact set implies approximability on the entire compact set. 
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A. B. Zaitsev. Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in $\mathbb{R}^2$. Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 41-51. http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a4/

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