Universal Taylor series with respect to a prescribed subsequence

Abstract

For a holomorphic function $f$ in the open unit disc $\mathbb{D}$ and $\zeta\in\mathbb{D}$, $S_n(f,\zeta)$ denotes the $n$-th partial sum of the Taylor development of $f$ at $\zeta$. Given an increasing sequence of positive integers $\mu=(\mu_n)$, we consider the classes $\mathcal{U}(\mathbb{D},\zeta)$ (resp. $\mathcal{U}^{(\mu)}(\mathbb{D},\zeta)$) of such functions $f$ such that the partial sums $\{S_n(f,\zeta):n=1,2,\dots\}$ (resp. $\{S_{\mu_n}(f,\zeta):n=1,2,\dots\}$) approximate all polynomials uniformly on the compact sets $K\subset\{z\in\mathbb{C}:\vert z\vert\geq 1\}$ with connected complement. We show that these two classes of universal Taylor series coincide if and only if $\limsup_n\left(\frac{\mu_{n+1}}{\mu_n}\right)<+\infty$. In the same spirit, we prove that, for $\zeta\ne 0,$ we have the equality $\mathcal{U}^{(\mu)}(\mathbb{D},\zeta)=\mathcal{U}^{(\mu)}(\mathbb{D},0)$ if and only if $\limsup_n\left(\frac{\mu_{n+1}}{\mu_n}\right)<+\infty$. Finally we deal with the case of real universal Taylor series.