Universal partial sums of Taylor series as functions of the centre of expansion

ABSTRACT

V. Nestoridis conjectured that if Ω is a simply connected domain of $\mathbb{C}$ that does not contain 0 and $S\left(\mathrm{\Omega }\right)$ is the set of all functions $f\in \mathcal{H}\left(\mathrm{\Omega }\right)$ with the property that the set $\{T_n\left(f\right)\left(z\right)=\sum _{k=0}^n{f}^{\left(k\right)}\left(z\right)/k!(-z{)}^k:n=0,1,2,\dots \}$ is dense in $\mathcal{H}\left(\mathrm{\Omega }\right)$, then $S\left(\mathrm{\Omega }\right)$ is a dense $G_{\delta }$ set in $\mathcal{H}\left(\mathrm{\Omega }\right)$. We answer the conjecture in the affirmative in the special case where Ω is an open disc $D(z_0,r)$ that does not contain 0.

COMMUNICATED BY:

• S. Ivashkovich

KEYWORDS:

• Taylor expansion
• partial sums
• universal taylor series
• ostrowski-gaps
• baire's category theorem
• generic property

AMS SUBJECT CLASSIFICATIONS:

• 30K99
• 30K05

Acknowledgments

I would like to thank V. Nestoridis, E. Archer and G. Vasdekis for taking interest in this work. I would also like to thank E. Katsoprinakis and the anonymous referees for their suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme [grant agreement no. 639046].