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Density of the polynomials in Hardy and Bergman spaces of slit domains

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Arkiv för Matematik

Abstract

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that \(\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\}\) and with the property that the analytic polynomials are dense in the Bergman space \(\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma)\) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in \(H^{t}(\mathbb{D}\setminus\Gamma)\) ; improving upon a result in an earlier paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Arkansas, Fayetteville, AR, 72701, U.S.A.

    John R. Akeroyd

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  1. John R. Akeroyd
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Correspondence to John R. Akeroyd.

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Akeroyd, J.R. Density of the polynomials in Hardy and Bergman spaces of slit domains. Ark Mat 49, 1–16 (2011). https://doi.org/10.1007/s11512-009-0110-8

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  • DOI: https://doi.org/10.1007/s11512-009-0110-8

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