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Families of Universal Taylor Series Depending on a Parameter

  • Evgeny Abakumov
  • Jürgen Müller
  • Vassili Nestoridis
Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

We construct families of universal Taylor series on Ω depending on a parameter w ∈ G, where Ω and G are planar simply connected domains. The functions to be approximated depend on the parameter w, w ∈ G. The partial sums implementing the universal approximation are one variable partial sums with respect to z ∈ Ω for each fixed value of the parameter w ∈ G. The universal approximation extends to mixed partial derivatives. This phenomenon is generic in H( Ω × G).

Keywords

Universal Taylor series Baire’s Theorem Runge’s Theorem Generic property Mixed partial derivatives 

A.M.S. Classification:

Primary 30K05 Secondary 32A30 

Notes

Acknowledgements

The authors are grateful to the referee for a careful reading of the manuscript and for his comments, which helped to improve the presentation considerably. Moreover, the authors thank N. Daras for a helpful communication. The work was supported by the Russian Science Foundation (Grant No. 14-41-00010).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Evgeny Abakumov
    • 1
    • 2
  • Jürgen Müller
    • 3
  • Vassili Nestoridis
    • 4
  1. 1.LAMA (UMR CNRS 8050)Université Paris-EstMarne-la-ValléeFrance
  2. 2.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  3. 3.University of TrierFB IV, MathematicsTrierGermany
  4. 4.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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