Approximation by harmonic functions

For a compact set X ⊂ ℝⁿ we construct a restoring covering for the space h(X) of real-valued functions on X which can be uniformly approximated by harmonic functions. Functions from h(X) restricted to an element Y of this covering possess some analytic properties. In particular, every nonnegative function f ∈ h(Y), equal to 0 on an open non-void set, is equal to 0 on Y. Moreover, when n = 2, the algebra H(Y) of complex-valued functions on Y which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set X ⊂ ℂ has a nontrivial Jensen measure, then X contains a nontrivial compact set Y with analytic algebra H(Y).