Geodesic
Uniform approximation by harmonic functions on compact subsets of~$\mathbb R^3$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 162-190

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We consider uniform approximation by harmonic functions on compact subsets in $\mathbb R^3$. Under an additional assumption that an approximated function is Dini-continuous, we prove a natural analog of Vitushkin's well-known uniform approximation lemma for an individual analytic function. 
@article{ZNSL_2011_389_a9,
     author = {M. Ya. Mazalov},
     title = {Uniform approximation by harmonic functions on compact subsets of~$\mathbb R^3$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {162--190},
     publisher = {mathdoc},
     volume = {389},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a9/}
}
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M. Ya. Mazalov. Uniform approximation by harmonic functions on compact subsets of~$\mathbb R^3$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 162-190. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a9/