关于给定子列的泛泰勒级数
Universal Taylor series with respect to a prescribed subsequence
摘要：
对于开单位圆盘$\mathbb{D}$和$ζ\in\mathbb{D}$中的全纯函数$f$，$S_n(f,ζ)$表示$f$在$ζ$处的泰勒展开的$n$次部分和。给定一个正整数递增序列$μ=(μ_n)$，我们考虑类$\mathcal{U}(\mathbb{D},ζ)$（分别为. $\mathcal{U}^{(μ)}(\mathbb{D},ζ)$），使得部分和$\{S_n(f,ζ):n=1,2,\dots\}$（分别$\{S_{μ_n}(f,ζ):n=1,2,\dots\}$）在具有连通补的紧集$K\subset\{z\in\mathbb{C}:\vert z\vert\geq 1\}$上一致逼近所有多项式.证明了这两类泛泰勒级数重合的充要条件是$\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$。在同样的精神，我们证明，对于$ζ\ne 0,$，我们有平等$\mathcal{U}^{(μ)}(\mathbb{D},ζ)=\mathcal{U}^{(μ)}(\mathbb{D},0)$当且仅当$\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$。最后讨论了实泛泰勒级数的情形。
英文摘要：
For a holomorphic function $f$ in the open unit disc $\mathbb{D}$ and $ζ\in\mathbb{D}$, $S_n(f,ζ)$ denotes the $n$-th partial sum of the Taylor development of $f$ at $ζ$. Given an increasing sequence of positive integers $μ=(μ_n)$, we consider the classes $\mathcal{U}(\mathbb{D},ζ)$ (resp. $\mathcal{U}^{(μ)}(\mathbb{D},ζ)$) of such functions $f$ such that the partial sums $\{S_n(f,ζ):n=1,2,\dots\}$ (resp. $\{S_{μ_n}(f,ζ):n=1,2,\dots\}$) 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 $\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$. In the same spirit, we prove that, for $ζ\ne 0,$ we have the equality $\mathcal{U}^{(μ)}(\mathbb{D},ζ)=\mathcal{U}^{(μ)}(\mathbb{D},0)$ if and only if $\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$. Finally we deal with the case of real universal Taylor series.
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