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Search: MSC category 32A05 ( Power series, series of functions )

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1. CMB 2011 (vol 54 pp. 230)

Clouâtre, Raphaël
Universal Power Series in $\mathbb{C}^N$
We establish the existence of power series in $\mathbb{C}^N$ with the property that the subsequences of the sequence of partial sums uniformly approach any holomorphic function on any well chosen compact subset outside the set of convergence of the series. We also show that, in a certain sense, most series enjoy this property.

Categories:32A05, 32E30

2. CMB 2006 (vol 49 pp. 256)

Neelon, Tejinder
A Bernstein--Walsh Type Inequality and Applications
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
Categories:32A05, 26E05

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