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Results & Lemmas (3)
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Lemma 1
Lemma 1. There exists a bounded, increasing function on T such that whenever E is a finite union of closed subintervals of T.
Lemma 1. There exists a bounded, increasing function $\beta$ on T such that
$$\lim_{n\to\infty}\int_E |\phi_n| = \int_E d\beta$$
whenever E is a finite union of closed subintervals of T.
Lemma 2
Lemma 2. There exists a sequence of sets; a sequence of positive numbers; and an increasing sequence of positive integers such that: - (a)…
Lemma 2. There exists a sequence of sets $E_1, E_2, \dots$ ; a sequence of positive numbers $\delta_1, \delta_2, \dots$ ; and an increasing sequence of positive integers $j_1, j_2, \dots$ such that:
- (a) Each $E_n$ is a finite union of closed intervals.
- (b) Let $E'_j$ denote the closure of the $\delta_j$ neighborhood of $E_j$ . Then $E'_j \subset G$ .
- (c) $j \neq k \Rightarrow E'_j \cap E'_k = \emptyset$ . [Note that this $\Rightarrow \lambda(E_j) \rightarrow 0$ , and $\lambda(E'_j) \rightarrow 0$ .]
- (d) $\int_{g-E_k} |\phi_{j_k}| < \varepsilon_0/2$ for $k=1,\,2,\,\cdots$
- (e) $\int_{E_n} (f-f_r)\phi_{j_n} \rightarrow x_0 \ where \ |x_0-L(f)| < 2\varepsilon_0.$
Lemma 3
Lemma 3. Let E be a closed subset of T,. Then there is a function, s, analytic on D such that: - (a) s has positive real part and - (b) -…
Lemma 3. Let E be a closed subset of T, $\varepsilon > 0$ . Then there is a function, s, analytic on D such that:
- (a) s has positive real part and $|s|_{\infty} < 1$
- (b) $\theta \in E \Longrightarrow |s(e^{i\theta}) 1| < \varepsilon$
- (c) $\theta \notin E \Longrightarrow |s(e^{i\theta})| < 2\lambda(E)/\varepsilon \cdot \operatorname{dist}(\theta, E)$
- (d) $|s(0)| < \lambda(E)/\varepsilon$ .