Bounded limits of analytic functions

Abstract

Let U be a bounded open plane set, and let / be a bounded analytic function on U, which is the pointwise limit of a bounded sequence {fn} of uniformly continuous analytic functions. It is shown that one can find another such sequence {fn}, converging to f, and bounded by the supremum norm off. A similar result is proved for approximation by rational functions. In this paper the following problem is considered: let U be a bounded open subset of the complex plane C, and let A be a set of bounded analytic functions on U. Which bounded analytic functions on U are limits of bounded sequences of functions in A converging pointwise in U? In the case where A consists of the polynomials this question was settled by Rubel and Shields [4], a special case had earlier been treated by Farrell [2]. A function f on U is the pointwise limit of some bounded sequence of polynomials if and only if it is the restriction to U of a bounded analytic function on U*, the Caratheodory hull of U. (U* is the interior of the complement of the unbounded component of the complement of the closure of U.) Rubel and Shields gave an example to show that the set of such pointwise limits need not be closed under uniform convergence. They constructed a set U and a sequence {fn} of bounded analytic functions converging uniformly on U tof, such that eachf, is a pointwise bounded limit of polynomials, but that the bounds on the approximating sequences of polynomials necessarily tended to infinity with n, so that no bounded diagonal subsequence converging to f could be found. The object of this paper is to show that if A is either the algebra of uniformly continuous analytic functions on U or (under mild hypotheses on aU) the rational functions with poles outside U, then this phenomenon cannot occur. The author is very grateful to T. W. Gamelin and B. K. 0ksendal for many valuable conversations. Notation. A(U) denotes the algebra of uniformly continuous analytic functions on the bounded open set U (which we regard as continuous Received by the editors March 26, 1971. AMS 1970 subject classifications. Primary 30A82; Secondary 30A98, 46J15.