Results & Lemmas (3)
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LEMMA 1.
LEMMA 1. There exists a bounded, increasing function β on T such that limί φn = dβ n-*oo JE JE whenever E is a finite union of closed…
LEMMA 1. There exists a bounded, increasing function β on T such that limί \φn\ = \ dβ n-*oo JE JE whenever E is a finite union of closed subintervals of T.
LEMMA 2.
LEMMA 2. There exists a sequence of sets El9 E2, a sequence of positive numbers δl9 δ2, and an increasing sequence of positive integers j,…
LEMMA 2. There exists a sequence of sets El9 E2, a sequence of positive numbers δl9 δ2, and an increasing sequence of positive integers j\, j2, such that: (a) Each En is a finite union of closed intervals. (b) Let Ej denote the closure of the δj neighborhood of Eά. Then E]dG. (c) jΦk^E'jΠEί^ 0. [Note that this =>x(Ej)—0, and o.]
LEMMA 3.
LEMMA 3. Let E be a closed subset of T, ε > 0. Then there is a function, s, analytic on D such that: (a) 8 has positive real part and [ s ^…
LEMMA 3. Let E be a closed subset of T, ε > 0. Then there is a function, s, analytic on D such that: (a) 8 has positive real part and [ s ^ < 1 (b) θeE=>\s(e iθ) - 1\ < ε (c) θ g E => \s{e iθ) I < 2X{E)/ε dist (θ, E) (d) \s(0)\<X(E)/ε.