Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains $$\Omega _i$$ Ω i that extend continuously on $$\prod \nolimits _{i=1}^{d}(\Omega _i \cup S_i)$$ ∏ i = 1 d ( Ω i ∪ S i ) , where $$S_i$$ S i are subsets of $$\partial \,\Omega _i$$ ∂ Ω i , open in the relative topology. The universal approximation occurs on every product of compact sets $$K_i$$ K i such that $$C-K_i$$ C - K i are connected and for some $$i_0$$ i 0 it holds $$K_{i_0}\cap (\Omega _{i_0}\cup \overline{S_{i_0}})=\varnothing $$ K i 0 ∩ ( Ω i 0 ∪ S i 0 ¯ ) = ∅ .

中文翻译：

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单连通域积上的部分光滑的通用泰勒级数

使用最近的 Mergelyan 类型定理，我们证明了在 $$\prod \nolimits _{i=1}^{d 上连续扩展的平面单连通域 $$\Omega _i$$ Ω i 的乘积上存在通用泰勒级数}(\Omega _i \cup S_i)$$ ∏ i = 1 d ( Ω i ∪ S i ) ，其中$$S_i$$ S i 是$$\partial \,\Omega _i$$ ∂ Ω i 的子集，在相对拓扑中打开。通用逼近出现在紧致集合 $$K_i$$ K i 的每个乘积上，使得 $$C-K_i$$ C - K i 是连通的，并且对于某些 $$i_0$$ i 0 它持有 $$K_{i_0} \cap (\Omega _{i_0}\cup \overline{S_{i_0}})=\varnothing $$ K i 0 ∩ ( Ω i 0 ∪ S i 0 ¯ ) = ∅ .