Let D be a bounded open
subset of the complex plane ⊄ which is the interior of its closure, and let h be a
bounded analytic function on D. The classical theorem of Runge implies that
there is a sequence of rational functions with poles in the complement of the
closure of D which converges to h uniformly on compact subsets of D. The
question naturally arises as whether this sequence may be chosen so that
the supremum norms over D of the rational functions remain uniformly
bounded. Of course, if the boundary of D consists of a finite number of
disjoint circles (that is, D is a circle domain), then it is a classical result
that the approximating sequence may be chosen so that their norms do
not exceed the norm of h. But suppose that the boundary of D is quite
complicated or D has infinitely many components in its complement. This general
question has been the subject of several recent papers and is the subject of this
one.
