BOUNDED APPROXIMATION BY RATIONAL FUNCTIONS

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. © 1969 by Pacific Journal of Mathematics.