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

Keywords:

Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series

MSC Classifications:

32A05, 26E05 show english descriptions Power series, series of functions
Real-analytic functions [See also 32B05, 32C05] 32A05 - Power series, series of functions
26E05 - Real-analytic functions [See also 32B05, 32C05]

top of page | contact us | social media | privacy | site map