Bounded approximation by rational functions

Abstract

In which it is proved that H~OL Q ) n C(K) = R(K).l Introduction* Let if be a compact subset of the complex plane C, and let R(K) be the uniform closure in C{K) of the rational functions with poles off K. Denote by Q the set of nonpeak points of R(K), let λ ρ be the area measure λ = dxdy restricted to Q, and let H°°(X Q ) be the weak-star closure of R(K) in L°°(X Q ).Our first goal is to prove the following theorem.As an immediate consequence, we can state the following more concrete version of the result: If fe C(K), and if there is a bounded sequence f n in R{K) converging pointwise almost everywhere {dxdy) to/, then/ejβ(2f).Theorem l l will be essentially a corollary of the following theorem of A. M. Davie [5].DA VIE 'S THEOREM.If /eiP°(λ ρ ), there is a sequence f n in R(K) such that H/JI ^ [|/||, and f n {q) ->f(q) for almost all (dxdy) points qeQ.Actually, Davie states explicitly the above result only for those feH°°(XQ) which are weak-star limits of bounded sequences in R(K).It follows then from the Krein-Schmulian theorem that the space of such functions is weak-star closed, and so must coincide with H~(X Q ).We will offer three proofs of Theorem 1.1.The first proof, given in §3, depends also on Vitushkin's description of the functions in R(K).In § §4 and 5, we present two "abstract" proofs of Theorem 1.1, which are dual to each other.All three proofs use a localization procedure.In § §6 and 7, some extensions of Theorem 1.1 are obtained by the methods of the abstract proof.These are pursued in the setting of uniform algebras.The first extension is a qualitative form of Theorem 1.1, which is inspired by a theorem of Sarason ([12]; see also [16], [6]).If he C(K), then the distance from h to R{K) is defined by d{h, R{K)) -inf {| | Λ -f\\:feR(K)} , and the distance d(h, H°°(X Q )) is defined similarly.The results of § 6, 129 130 T. W. GAMELIN AND JOHN GARNETT applied to R(K), will yield the following theorem, which has Theorem 1.1 as an immediate consequence.THEOREM 1.2.d(h, R{K)) = d(h, ίP°(λ ρ )) for all heC(K).Equivalently, ball [R{K) L Π L X (X Q )] is weak-star dense in ball R(K) L .B. Cole had observed (unpublished) that a result like Davie's theorem could be used to show that every nonpeak point q e Q has a representing measure which is absolutely continuous with respect to X Q .The results of §7, again specialized to the algebra R(K), include the following extension of Cole's result.THEOREM 1.3.For each q e Q, the representing measures for q which are absolutely continuous with respect to X Q are weak-star dense in the set of all representing measures for q.

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF