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\cl{Lecture on Harmonic Approximation}

\cl{Stephen J. Gardiner}
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\noindent
{\bfb Abstract}:
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 $\Bbb C$, 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
$\Omega\setminus K$ 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 $\Bbb R^n$ ($n\geq2$).

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 $\Bbb R^n$ 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 $\Bbb R^n\setminus K$  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 $\Bbb R^n$: 
if $(\Bbb R^n\cup\{\infty\})\setminus E$ 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 $\Bbb R^n$. 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 $f:\, \Bbb R \rightarrow 
\Bbb C$ and   $\varepsilon :\, \Bbb R \rightarrow (0,1]$, there exists
an entire function $g$ such that $|g-f|\leq \varepsilon $  on $\Bbb R$.

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.

\bye 

