Bounded point evaluations for certain polynomial and rational modules

Abstract

Let K be a compact subset of the complex plane C.

Reproducing kernel of the space $R^t(K,μ)$

“…Let ǫ 1 be chosen as in Lemma 2 of [14]. By our assumption γ(D \ K) < ǫ 1 and Lemma 2 of [14], we conclude that Case II on Page 225 of [14] holds, that is, scheme(Q, ǫ, m, γ n , Γ n , n ≥ m) (ǫ < 10 −3 ) does not terminate. In this case, one has a sequence of heavy ǫ barriers inside Q, that is, {γ n } n≥m and {Γ n } n≥m are infinite.…”

“…Proof. We use Thomson's coloring scheme that is described at the beginning of section 2 of [14]. Let ǫ 1 be chosen as in Lemma 2 of [14].…”

“…We use Thomson's coloring scheme that is described at the beginning of section 2 of [14]. Let ǫ 1 be chosen as in Lemma 2 of [14]. By our assumption γ(D \ K) < ǫ 1 and Lemma 2 of [14], we conclude that Case II on Page 225 of [14] holds, that is, scheme(Q, ǫ, m, γ n , Γ n , n ≥ m) (ǫ < 10 −3 ) does not terminate.…”

“…Let f ∈ A(D), by the maximal modulus principle, we can find z n ∈ γ n such that |f (λ)| ≤ |f (z n )| for λ ∈ R. By the definition of γ n , we can find a heavy ǫ square S n with z n ∈ S n ∩ γ n . Since γ(Int(S n ) \ K) ≤ ǫd Sn (see (2.2) in [14]), we must have Area(S n ∩ K) > 0. We can choose…”