全纯系数形式幂级数的收敛集  

On Convergence Sets of Power Series with Holomorphic Coefficients

作　　者：刘华 Basma Al-Shutnawi Liu Hua;Basma Al-Shutnawi(Shanghai Technical Institute of Electronics and Information,Shanghai 201411;Department of Mathematics,Tafila Technical University,Tafila 661109)

机构地区：[1]上海电子信息职业技术学院,上海201411 [2]Tafila Technical University Tafila,(约旦)661109

出　　处：《数学物理学报（A辑）》2024年第3期563-574,共12页Acta Mathematica Scientia

摘　　要：该文研究形式幂级数f(z,t)=∑_(n=0)^(∞)fn(z)t^(n)的收敛集,这里系数fn(z)是复平面上某个域Ω上的全纯函数.Ω的一个子集E被称为Ω上的收敛集,如果存在形式幂级数f(z,t)使得E恰好包含使得f(z,t)作为t的幂级数在原点的某个邻域内收敛的所有z.σ-凸集被定义为可数个多项式紧凸子集的并.证明了复平面的子集是收敛集当且仅当它是σ-凸的.We consider convergence sets of formal power series f(z,t)=∑_(n=0)^(∞)fn(z)t^(n),where fn(z)are holomorphic functions on a domainΩin C.A subset E ofΩis said to be a convergence set inΩif there is a series f(z,t)such that E is exactly the set of points z for which f(z,t)converges as a power series in a single variable t in some neighborhood of the origin.Aσ-convex set is defined to be the union of a countable collection of polynomially convex compact subsets.We prove that a subset of C is a convergence set if and only if it isσ-convex.

关 键 词：形式幂级数 解析函数 收敛集 

分 类 号：O174.56[理学—数学]