Universal Faber and Taylor series on an unbounded domain of infinite connectivity

Abstract

We give the first example of a Taylor (or Faber) universal series on an unbounded domain in ℂ whose complement has an infinite, even non-denumerable, set of components. This extends the previous results of Costakis, Vlachou, Melas and Bacharoglou.

Keywords:

• Taylor series
• Faber series
• universal function
• Runge's approximation theorem
• Baire's Theorem
• algebraic genericity

AMS Subject Classifications::

• 30B30
• 30E10

Acknowledgements

This article has been written after I read 10 and I followed a seminar talk of Prof. Vagia Vlachou related to 17. I am grateful to Prof. Vassili Nestoridis for helpful comments and suggestions and his interest in this work. I would also like to thank the anonymous referees for their significant help in the presentation of this work. I also express my gratitude to Prof. Stephen Gardiner for his interest in this work.