Approximation by harmonic functions

Poletsky, Evgeny

doi:10.1090/S0002-9947-97-02041-2

Approximation by harmonic functions
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by Evgeny A. Poletsky PDF

Trans. Amer. Math. Soc. 349 (1997), 4415-4427 Request permission

Abstract:

For a compact set $X\subset \mathbb R^n$ 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 $f\in h(Y)$, 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 $X\subset \mathbb C$ has a nontrivial Jensen measure, then $X$ contains a nontrivial compact set $Y$ with analytic algebra $H(Y)$.

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