Results in Mathematics

, Volume 29, Issue 1–2, pp 1–15 | Cite as

Recent Developments In Harmonic Approximation, With Applications

  • D. H. Armitage
  • P. M. Gauthier
Article

Abstract

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rn with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rn. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.

1991 Mathematics Subject Classification

30E10 31-02 31B05 41-02 41A63 

Key words and phrases

Harmonic functions approximation universal functions Radon transform maximum principle Dirichlet problem 

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Copyright information

© Birkhäuser Verlag, Basel 1996

Authors and Affiliations

  • D. H. Armitage
    • 1
  • P. M. Gauthier
    • 2
  1. 1.Department Of Pure MathematicsQueen’s UniversityBelfast BT7 InnNorthern Ireland
  2. 2.Départment De Mathématiques Et De StatistiqueUniversité De MontréalMontréalCanada

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