Approximation by harmonic functions

Author:
Evgeny A. Poletsky

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4415-4427

MSC (1991):
Primary 32F05; Secondary 32E25, 32E20

DOI:
https://doi.org/10.1090/S0002-9947-97-02041-2

MathSciNet review:
1443888

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

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DOI:
https://doi.org/10.1090/S0002-9947-97-02041-2

Keywords:
Harmonic functions,
potential theory,
uniform algebras

Received by editor(s):
September 10, 1995

Article copyright:
© Copyright 1997
American Mathematical Society