The Structure of the Closure of the Rational Functions in Lq(μ)

Abstract

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K • . Let µ be a positive finite measure with its support contained in K.. The aim of this work is to study the structure of the space A q (K, µ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A q (K, µ) can be established. Our theorem deduces J. Thomson's structure theorem for P q (µ), the closure of polynomials in L q (µ), as the special case when K is a closed disk containing the support of µ. Mathematics Subject Classification (2000). Primary 46E30; Secondary 30H05, 30E10, 46E15.

Carleson measures on planar sets

“…Since the restrictions of ω 1 to the boundaries of the components of V \ ∂G are not mutually singular, we see that ∇A q (V, ω 1 ) = V \ ∂G. So ∇A q (V, ω 1 ) is a simply connected domain such that V \ ∂G ⊂ ∇A q (V, ω 1 ) ⊂ V. According to Lemma 3 in [15] (or see the main theorem in [16]), ∇A q (V, ω 1 ) must be nicely connected. Since V ∩ ∂G is a smooth Jordan arc, it follows that ∇A q (V, ω 1 ) = V .…”

Beurling’s theorem and invariant subspaces for the shift on Hardy spaces

“…We use ∇R q (K, µ) to denote the set of abpes for R q (K, µ). The proof of our main theorem needs the following structure theorem for R q (K, µ), which can be found in [16]. Recall that the connectivity of a connected domain is defined to be the number of the components of its complement in the complex plane.…”