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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Farrell sets for harmonic functions
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by Stephen J. Gardiner and Mary Hanley PDF
Proc. Amer. Math. Soc. 131 (2003), 773-779 Request permission

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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Additional Information
  • Stephen J. Gardiner
  • Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
  • MR Author ID: 71385
  • ORCID: 0000-0002-4207-8370
  • Email: stephen.gardiner@ucd.ie
  • Mary Hanley
  • Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
  • Email: mary.hanley@ucd.ie
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 773-779
  • MSC (2000): Primary 31B05; Secondary 41A28
  • DOI: https://doi.org/10.1090/S0002-9939-02-06776-X
  • MathSciNet review: 1937416