Smooth Universal Taylor Series

Abstract.

Let $\Omega \subseteq {\Bbb C}$ be a simply connected domain in ${\Bbb C}$, such that $\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}]$ is connected. If g is holomorphic in Ω and every derivative of g extends continuously on $\bar{\Omega}$, then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and $\zeta \in \bar{\Omega}$ we denote $S_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l$. We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

• i) There exists a strictly increasing sequence μ_n ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ $\bar{\Omega}$ and every l ∈ {0, 1, 2, …} we have $\sup_{\zeta \in \Gamma} \sup_{w \in \Delta} \frac{\partial^l}{\partial w^l} S_{\mu_ n} (\,f,\zeta) (w)-f^{(l)}(w) \rightarrow 0, \hskip 7.8pt {\rm as}\,n \rightarrow + \infty \quad {\rm and}$

• ii) For every compact set $K \subset {\Bbb C}$ with $K\cap \bar{\Omega} =\emptyset$ and K^c connected and every function $h: K\rightarrow {\Bbb C}$ continuous on K and holomorphic in K⁰, there exists a subsequence $\{ \mu^\prime _n \}^{\infty}_{n=1}$ of $\{\mu_n \}^{\infty}_{n=1}$, such that, for every compact set $L \subset \bar{\Omega}$ we have $\sup_{\zeta \in L} \sup_{z\in K} S_{\mu^\prime _n} (\,f,\zeta )(z)-h(z) \rightarrow 0, \hskip 7.8pt {\rm as} \, n\rightarrow + \infty .$