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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Farrell sets for harmonic functions


Authors: Stephen J. Gardiner and Mary Hanley
Journal: Proc. Amer. Math. Soc. 131 (2003), 773-779
MSC (2000): Primary 31B05; Secondary 41A28
Published electronically: September 17, 2002
MathSciNet review: 1937416
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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 $\left\vert h\right\vert \leq M$ 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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Additional Information

Stephen J. Gardiner
Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
Email: stephen.gardiner@ucd.ie

Mary Hanley
Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
Email: mary.hanley@ucd.ie

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06776-X
PII: S 0002-9939(02)06776-X
Received by editor(s): April 18, 2001
Received by editor(s) in revised form: October 10, 2001
Published electronically: September 17, 2002
Additional Notes: This research was partially supported by EU Research Training Network HPRN-CT-2000-00116
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2002 American Mathematical Society