On a joint approximation question in \(H^p\) spaces

Abstract

We consider Mergelyan sets and Farrell sets for \(H^p\)\((1\le p < \infty )\) spaces in the unit disc for both the weak topology and the norm topology, and give a short proof of a theorem of Pérez-González which answers a question proposed by Rubel and Stray (J Approx Theory 37:44–50, 1983).

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Notes

  1. 1.

    The formulation of the Banach–Saks theorem in [8, p. 80], does not specify an interval for p, but the interval is \(1<p<\infty \) (see, for example, [1, p. 52]). Obviously in [8, p. 80], it is assumed that the restriction on p is the same as in the formulation of the previous theorem in [8, p. 78].

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Acknowledgements

The author is grateful to the referee for careful reading of the manuscript and corrections.

Funding

Funding was provided by Simons Foundation (US) (Grant No. 430329).

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Correspondence to Arthur A. Danielyan.

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Danielyan, A.A. On a joint approximation question in \(H^p\) spaces. Arch. Math. (2020). https://doi.org/10.1007/s00013-020-01478-9

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Keywords

  • Mergelyan set
  • Farrell set
  • \(H^p\) spaces
  • Joint approximation by polynomials

Mathematics Subject Classification

  • Primary 30E10
  • 46E15