, Volume 29, Issue 1-2, pp 1-15
Date: 17 Apr 2013

Recent Developments In Harmonic Approximation, With Applications

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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.

Dedicated to the memory of our friend and colleague, Myron Goldstein, 1935–94.