论文标题
通用泰勒系列有关规定的子序列
Universal Taylor series with respect to a prescribed subsequence
论文作者
论文摘要
对于holomorthic函数$ f $ in Open Unit Disc $ \ Mathbb {D} $和$ζ\ in \ Mathbb {d} $，$ s_n（f，ζ）$表示$ f $ $ f $ at $ζ$ $ f $的$ n $ th $ n $ thaul-thoptial-th $ n $ the。给定越来越多的正整数顺序$μ=（μ_n）$，我们考虑类$ \ nathcal {u}（\ Mathbb {d}，ζ）$（wesp。 $ \ {s_n（f，ζ）：n = 1,2，\ dots \} $（resp。$ \ {s_ {s_ {μ_n}（f，ζ）：n = 1,2，\ dots \} $）近似于所有多项式在compact $ k \ k \ sebset \ sebset \ sex z \ in z z \ in c \ in c \ in s sem ar of All dolynomials上Z \ vert \ geq 1 \} $具有连接的补充。我们表明，当且仅当$ \ limsup_n \ left（\ frac {μ_{n+1}}} {μ_n} \ right）<+\ infty $时。本着同样的精神，我们证明，对于$ζ\ ne 0，$，我们具有平等$ \ Mathcal {u}^{（μ）}（\ Mathbb {d}，ζ）= \ Mathcal {u}^{u}^{（μ）}（μ）}（\ Mathbb {\ Mathbb {d}，if和if if和if if和if if和if if和if If if if和if If If if if和if If If if和if $ \ limsup_n \ left（\ frac {μ_{n+1}}} {μ_n} \ right）<+\ infty $。最终，我们处理了真正的通用泰勒系列案例。
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.
