### Abstract

For a compact set X ⊂ 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 ∈ 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).

Original language | English (US) |
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Pages (from-to) | 4415-4427 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 349 |

Issue number | 11 |

State | Published - Dec 1 1997 |

### Keywords

- Harmonic functions
- Potential theory
- Uniform algebras

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*349*(11), 4415-4427.