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The year 1885 has a special significance in the history of
approximation theory. It was then that Weierstrass published
his famous result which says that a continuous function on a closed
bounded interval can be uniformly approximated by polynomials.
The same year saw the birth of holomorphic approximation in the
celebrated paper of Runge. Given an open set Omega in the
complex plane C, which compact subsets K have the property
that any holomorpic function defined on a neighbourhood of K can
be uniformly approximated on K by functions holomorphic on Omega ?
Runge's theorem supplies the answer: precisely the sets K such that
Omega - K has no components which are relatively compact
in Omega. Since Runge's original work holomorphic approximation
has developed into an active research area. We mention
particulary the contribution of Carleman, Alice Roth, Mergelyan,
Arakelyan and Nersesyan. A helpful account of these and other results
can be found in the lecture notes by Gaier (Lectures on Complex
Approximation, Birkh„user, 1987). The purpose of these lectures is to
give a corresponding account of the theory of approximation by harmonic
function in Euclidean space R^n (n>=2).
The starting point in the history of harmonic approximation is not as easy to identify. In the case of approximation in higher dimensions, a paper of J.L. Walsh in 1929 seems a reasonable choice, but for approximation in the plane mention must also be made of work of Lebesgue in 1907. Which compact sets K in R^n have the property that any harmonic function defined on a neighbourhood of K can be uniformly approximated on K by harmonic polynomials? Walsh's Theorem tells us that, if R^n - K is connected, then such approximation is always possible. However, unlike the case of holomorphic approximation, the converse to this statement is false. The characterization of compact sets K with the above approximation property is rather more delicate than in the case of Runge's Theorem, and involves the potential theoretic notion of "thin sets".
Until comparatively recently most of the work was in terms of approximation on compact sets. This changed in the early 1980's due to progress made by Gauthier, Goldstein and Ow. Inspired by work of Alice Roth in the holomorphic case, they developed a technique of "fusing" two harmonic functions which are close in value on a certain set. As a result they obtained, in particular, a generalization of Walsh's Theorem to the case of approximation on closed (but not necessarily bounded) sets E in R^n : if (R^n u {\infty}) - E is connected and locally connected, then functions harmonic on an open set containing E can be uniformly approximated on E by functions harmonic on all of R^n. As was the case with Walsh's Theorem, the above hypotheses concerning connectedness are sufficient, but not necessary for this type of approximation to be possible. A complete characterization of sets E which posses this approximation property has been obtained recently. Indeed, a period of rapid development has brought a new coherence and substance to the whole subject of approximation by harmonic functions. These lectures will give an organised account of harmonic approximation which includes several of these new results.
The plan is roughly as follows. We will assume a basic knowledge of potential theory as far as the Dirichlet problem, but will not assume familiarity with thin sets and the fine topology. Uniform harmonic approximation on compact sets, and then on relatively closed sets, will be developed by describing the relevant techniques of pole-pushing and fusion. We then turn our atention to the question of better-then-uniform approximation: that is, can it be arranged that the error in our approximation decays to 0 as we approach "infinity"? In the holomorphic case one such famous result is due to Carleman. He showed that, given any continuous functions f:R->C and epsilon:R->(0,1], there exists an entire function g such that |g-f|<=epsilon on R.
Finally, one of the most rewarding aspects of the whole subject is its potential for applications, sometimes surprising ones. A selection of these applications will be presented, dealing with (in particular) the Dirichlet problem for unbounded regions, domains (possibly unbounded) for which the maximum principle holds, a non-uniqueness property of the Radon transform, and boundary behaviour of harmonic and superharmonic functions.