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

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 $\left\{T_N(f)(z):={\sum }_{n=0}^N\frac{f^{(n)}(z)}{n!}(-z{)}^n:N=0,1,2,\dots \right\}$ 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 $0$.

9 pages