Results in Mathematics

, Volume 29, Issue 1, pp 1–15

Recent Developments In Harmonic Approximation, With Applications

  • D. H. Armitage
  • P. M. Gauthier

DOI: 10.1007/BF03322201

Cite this article as:
Armitage, D.H. & Gauthier, P.M. Results. Math. (1996) 29: 1. doi:10.1007/BF03322201


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


Key words and phrases

Harmonic functionsapproximationuniversal functionsRadon transformmaximum principleDirichlet problem

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