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Existence of Universal Taylor Series for Nonsimply Connected Domains

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Abstract

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.

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Authors and Affiliations

  1. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

    Stephen J. Gardiner

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  1. Stephen J. Gardiner
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Correspondence to Stephen J. Gardiner.

Additional information

Communicated by Vilmos Totik.

This research was supported by Science Foundation Ireland under Grant 09/RFP/MTH2149, and is also part of the program of the ESF Network “Harmonic and Complex Analysis and Applications” (HCAA).

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Gardiner, S.J. Existence of Universal Taylor Series for Nonsimply Connected Domains. Constr Approx 35, 245–257 (2012). https://doi.org/10.1007/s00365-011-9133-z

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  • DOI: https://doi.org/10.1007/s00365-011-9133-z

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