Pointwise bounded approximation by functions satisfying a side condition

Abstract

In this paper necessary and sufficient conditions on a subset S of the unit disc D are given such that every bounded analytic function / on D is a pointwise limit of a sequence {/nln=i of uniformly continuous analytic functions on D bounded by the sup norm of / and in addition satisfying sup{|/ n zeS} ^ sup{|/(z) I, zeS} for all n. 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 {z n}ζ =1 in D converging to ze T converges nontangen-tially to z if for some constant λ we have | z — z n | ^ λ(l — \\z n \\) 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(z n) 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 f