On countably universal series in the complex plane

ABSTRACT

Let $({\lambda }^k{)}_{k\ge 1}$ be countably many increasing sequences of integers tending to ∞. For a power series $f=\sum _i{a}_i{z}^i$, we denote by $S_n\left(f\right)=\sum _{i=0}^n{a}_i{z}^i$ the nth partial sum of f. Under some condition on $({\lambda }^k{)}_{k\ge 1}$, we construct lacunary power series converging in the unit disk $\mathbb{D}$, with arbitrarily large lacunes, which satisfy the following property (P): for any countable family $(K_k{)}_k$ of compact sets in the complement of $\mathbb{D}$, each of them with connected complement, the set $\left\{\right(S_{{\lambda }_n^1}\left(f\right),S_{{\lambda }_n^2}\left(f\right),\dots );n\in \mathbb{N}\}$ is dense in $A\left(K_1\right)\times A\left(K_2\right)\times A\left(K_3\right)\times \cdots$, endowed with the product topology, inherited from the $sup$-norm of the space $A\left(K\right)$ of all continuous functions on K which are holomorphic in its interior. The set of such countably universal series is shown to be invariant under some summability processes. Our construction also allows us to exhibit power series with Padé approximants enjoying countably universal properties, and to prove that the set of all power series with radius of convergence 1 which satisfy (P) is either void or spaceable. We finally use Ostrowski-gaps to give a very direct proof of two known results: (1) The set of doubly universal Taylor series is non-void only if $\frac{{\lambda }_{{\nu }_l}^1}{{\lambda }_{{\nu }_l}^2}\to \mathrm{\infty }\mathrm{or}\frac{{\lambda }_{{\nu }_l}^1}{{\lambda }_{{\nu }_l}^2}\to 0,l\to \mathrm{\infty },$ for some increasing sequence $({\nu }_l{)}_l$, and (2) The set of frequently universal series is void.

COMMUNICATED BY:

• M. Ruzhansky

KEYWORDS:

• Universal Taylor series
• Ostrowski-gaps
• Padé approximant

AMS SUBJECT CLASSIFICATIONS:

• 30K05
• 47A16

Acknowledgements

The author is grateful to the anonymous referee for her/his careful reading of the manuscript and valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This project was partly supported by the French National Research Agency (Agence Nationale de la Recherche, ANR) (project Front) [grant number ANR-17-CE40-0021].