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

TL;DR

This paper proves that for certain simply connected domains not containing zero, the set of functions with dense partial sums of their Taylor series centered at a point is a dense G_delta set, confirming a conjecture in a specific case.

Contribution

The paper confirms Nestoridis's conjecture for the case where the domain is an open disc not containing zero, establishing the density and G_delta property of the set of functions with dense partial sums.

Findings

Confirmed the conjecture for open discs not containing zero

Showed the set of functions with dense partial sums is dense in the function space

Established the set as a G_delta set in the space of holomorphic functions

Abstract

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