Pointwise bounded approximation by functions satisfying a side condition

Abstract

Let D = {z: | z | < 1} denote the open unit disc and T — {z: z | = 1} the unit circle. H°°(D) denotes all bounded analytic functions on D and A(D) consists of all uniformly continuous / in H°*(D). For any fe H°°{D) and any subset S of D we put || / | | s = sup {\f(z) |, zeS} and w e s e t 11/11 = \\f\\D. A sequence {zn}ζ=1 in D converging to ze T converges nontangentially to z if for some constant λ we have | z — zn | ^ λ(l — \zn\) for all n. JίfeH°°(D) then Fatou's theorem [2, page 34] tells us that / has a nontangential limit at almost every boundary point. Thus at almost every boundary point lim f(zn) exists and is independent of the choice of sequence. If fe H°°(D) is known a.e. on T we recapture its values in D by the Cauchy or Poisson integral formula. We will therefore consider functions in H°°(D) as defined in D and a.e. on T. A relatively closed subset S of D is called a Farrell set if for each fe H°°(D) there are fn e A, n = 1, 2, , converging pointwise to fonD with || Λ || ^ || / || and such that || fn \\s || f\\8. This concept was introduced to us by Professor L. A. Rubel who also raised the question of describing such sets. The object here is to characterize Farrell sets in terms of their cluster points on T. The author is very grateful to Dr. A. M. Da vie for valuable conversations on this subject. First we observe that if rze S whenever 0 < r < 1 and z e S, then S is a Farrell set. Indeed, letting fr{z) = f(rz)(0 < r <l), we have: fr(z) -*f(z) as r —* 1. On the other hand, let S = {zn}~=1 where Σ * 0— I z» I) < °° a n ^ assume that the set of cluster points of S on T has positive linear measure. Then there are fe H°°(D) with / = 0 on S, but / Φ 0, while if fe A(D) and / = 0 on S we must have / = 0. The set of cluster points of S on T which are not nontangential limits of sequences from S is here too large. In fact we will prove the following.