Archiv der Mathematik

, Volume 87, Issue 5, pp 427–435 | Cite as

Boundedness, regularity and smoothness of universal Taylor series



Let Ω be an unbounded simply connected domain in \(\mathbb{C}\) satisfying some topological assumptions; for example let Ω be an open half-plane. We show that there exists a bounded holomorphic function on Ω which extends continuously on \(\bar{\Omega}\) and is a universal Taylor series in Ω in the sense of Luh and Chui–Parnes with respect to any center. Our proof uses Arakeljan’s Approximation Theorem.

Further we strengthen results of G. Costakis [2] concerning universal Taylor series with respect to one center in the sense of Luh and Chui–Parnes in the complement G of a compact connected set. We prove that such functions can be smooth on the boundary of G and be zero at ∞. If the universal approximation is also valid on ∂G, then the function can not be smooth on ∂G, but it may vanish at ∞. Our results are generic in natural Fréchet spaces of holomorphic functions.

Mathematics Subject Classification (2000).



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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AthensAthensGreece

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