A STRONG NOTION OF UNIVERSAL TAYLOR SERIES

Abstract

For a holomorphic function $f$ in the open unit disc $D$, the $N$th partial sum of its Taylor series with center $\zeta \in D$ is denoted by $S_N(f,\zeta)(z)=$${\sum\nolimits^N_{n=0}}({{f^{(n)}(\zeta)}/n!})(z-\zeta)^n$. Generically, all functions $f$ in $H(D)$ satisfy the following. For every compact set $K\subset\Bbb C$ with $K{\cap}\,D=\varnothing$ and $K^c$ connected and every polynomial $h$, there exists a sequence of positive integers $\{\lambda_n\}^{\infty}_{n=1}$ such that, for every $l \in \{ 0,1,2, \ldots \}$, \[ \sup_{z \in K}\Big\vert {{\partial^l}\over {\partial z^l}} S_{\lambda_n}(f,0)(z)-h^{(l)}(z)\Big\vert \,{\to}\,0 \quad {\rm as} \; n\,{\to}\,{+}\,\,\infty.\]