A Bernstein--Walsh Type Inequality and Applications

Canadian mathematical bulletin

关键词: Bernstein-Walsh inequality;    convergence sets;    analytic functions;    ultradifferentiable functions;    formal power series;   

DOI  :  10.4153/CMB-2006-026-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

A Bernstein--Walsh type inequality for $C^{infty }$ functions of severalvariables is derived, which then is applied to obtain analogs andgeneralizations of the following classical theorems: (1) Bochnak--Siciaktheorem: a $C^{infty }$ function on $mathbb{R}^{n}$ that is realanalytic on every line is real analytic; (2) Zorn--Lelong theorem: if adouble power series $F(x,y)$ converges on a set of lines of positivecapacity then $F(x,y)$ is convergent; (3) Abhyankar--Moh--Sathaye theorem:the transfinite diameter of the convergence set of a divergent series iszero.

【 授权许可】

Unknown