Overconvergence of sequences of rational functions with sparse poles

Abstract

In this paper we consider Banach spaces S of functions holomorphic in D + = {z II z I< 1} which contain all functions {i/(z -b)) with b E D-= {z I[ z I > 1}.We show (Theorem 1) that under rather mild restrictions on S, a sequence {/~) of rational functions which converges in the norm of S, if the poles of all/n are confined to a "sparse" subset E of D-(here the sparseness criterion is determined by the particular space S, and we give it only in an implicit form), necessarily converges uniformly on compact subsets of D-\E.This it seems appropriate to call an "overconvergence" theorem, although it has a somewhat different character than other theorems bearing this designation, e.g.those due to Ostrowski [3] and to Walsh ([5], p. 77).In general the sequence {/~} will not converge in domains which intersect the unit circumference, and the limit functions in D + and D-are not analytic continuations of one another.In Theorem 2 it is shown, with additional hypotheses on S, that when the closure of E does not include the entire unit circumference we get overconvergence in domains which intersect the circumference, and hence analytic continuability of the limit functions.These theorems are proved in w 1.The present paper was motivated by a study of the paper [1] of Akutowicz and Carleson, from which we have adapted the method of proof of Theorem 2. Conversely, many of the theorems of [1] are deducible from our Theorem 2. In w 2 the relationship of our paper with [1] is discussed briefly.In w 3 some problems which invite further investigation are pointed out.

Locations

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