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 Aq(K, μ) denote the closure of A(K) in Lq(μ). The aim of this work is to study the structure of the space Aq(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for Aq(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for Pq(μ), the closure of polynomials in Lq(μ), as the special case when K is a closed disk containing the support of μ.
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Qiu, Z. The Structure of the Closure of the Rational Functions in Lq(μ). Integr. equ. oper. theory 59, 223–244 (2007). https://doi.org/10.1007/s00020-007-1514-0
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DOI: https://doi.org/10.1007/s00020-007-1514-0