6

L . Brown et al.

We shall now consider various plausible definitions of asymptotic tion. First of all, we repeat the definition we have been using so far.

Definition 1. H{U) is asymptotically dense in A{F) if for some г-gauge, H{U) is e-dense in A{F).

Notice that in this definition, the same e is used for every approximation. In the following, s may depend on /.

Definition 2. H{U) is asymptotically dense in A{F) if for each feA(F\ there is 3, g e H{U) such that

\f { z ) - g { z ) \ - ^0 , as IzHl, zeF.

In terms of applications to cluster sets, the following is also natural.

Definition 3. H{U) is asymptotically dense in A{F) if for each / e A{FX there is a g eH{U) such that

X ( / ( z ) , ^ ( z ) bO , as |zHl, zeF,

where x denotes a metric on the extended plane Cu{oo}.

Corollary 5. Suppose H{U) is uniformly dense in /1(F). Then all three definitions of asymptotic approximation are equivalent on F.

Proof . Clearly Definition l=>Definition 2=>Definition 3. Suppose now, H{U) is asymptotically dense in A{F) in the sense of Definition 3. As in the proof of Lemma 2, let a e U\F and

/ { z ) = (z-a)-^

Since / is bounded on F and H{U) is asymptotically dense in A{F) in the sense of Definition 3, there is a g e H{U) with

| / ( z ) - öf { z ) HO , as |zHl, zeF.

Now set h=f — g. Then h satisfies 3) in Theorem 3, and hence, H{U) is cally dense in the sense of Theorem 3, that is, in the sense of Definition 1.

3 . Approximation and Uniqueness

We have already hinted at the relation between approximation and uniqueness. In this section we summarize and elaborate this relation.

Definition 4. A closed set F in (7 is a set of asymptotic uniqueness if for all heH{Vl

[ / i ( z ) - ^0 , as |zH 1, z e F]=>[Ä = 0] .

Definition 5. A closed set F in [/ is a set of asymptotic approximation (by щи)) if H{U) is asymptotically dense in A{F).

Theorem 5. A closed set F in U is a set of asymptotic uniqueness if and only if F is not contained in a (proper) set of asymptotic approximation.

Proof . Suppose first F С F^ Ф C/, where F^ is a set of asymptotic approximation. Then by Theorem 3, Fj, and a fortiori F, is not a set of asymptotic uniqueness.