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. For 1 ≤ q < ∞, let A q (K, μ) denote the closure of A(K) in L q (μ). 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.