The paper is devoted to uniform approximation by harmonic functions on compact sets. The result is an approximation theorem for an individual function under the condition that, on the complement to the compact set, the harmonic capacity is "homogeneous" in a sense. The proof involves a refinement of Vitushkin's localization method. §0. Introduction Let X ⊂ R 3 be a compact set, X • the interior of X, Δ the Laplace operator in R 3 . We define two classes of functions h(X) and H(X) as follows:

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... H(X) is the closure in C(X) of the set {f | X : Δf = 0 in a neighborhood of X}