Approximation by harmonic functions

Abstract

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