Farrell sets for harmonic functions

Gardiner, Stephen; Hanley, Mary

doi:10.1090/S0002-9939-02-06776-X

Farrell sets for harmonic functions
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by Stephen J. Gardiner and Mary Hanley

Proc. Amer. Math. Soc. 131 (2003), 773-779

DOI: https://doi.org/10.1090/S0002-9939-02-06776-X

Published electronically: September 17, 2002

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Abstract:

Let $F$ denote a relatively closed subset of the unit ball $B$ of $\mathbb {R} ^{n}$. The purpose of this paper is to characterize those sets $F$ which have the following property: any harmonic function $h$ on $B$ which satisfies $\left | h\right | \leq M$ on $F$ (where $M>0$) can be locally uniformly approximated on $B$ by a sequence of harmonic polynomials which satisfy the same inequality on $F$. This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

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