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# A Bernstein--Walsh Type Inequality and Applications

Published:2006-06-01
Printed: Jun 2006
• Tejinder Neelon
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## Abstract

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
 MSC Classifications: 32A05 - Power series, series of functions 26E05 - Real-analytic functions [See also 32B05, 32C05]