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

John R. Akeroyd Department of Mathematics, University of Arkansas

Functional Analysis mathscidoc:1701.12014

Arkiv for Matematik, 49, (1), 1-16, 2009.3
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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@inproceedings{john2009density,
  title={Density of the polynomials in Hardy and Bergman spaces of slit domains},
  author={John R. Akeroyd},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203628865667990},
  booktitle={Arkiv for Matematik},
  volume={49},
  number={1},
  pages={1-16},
  year={2009},
}
John R. Akeroyd. Density of the polynomials in Hardy and Bergman spaces of slit domains. 2009. Vol. 49. In Arkiv for Matematik. pp.1-16. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203628865667990.
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