Recent Developments In Harmonic Approximation, With Applications
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- 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.