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On Borel summability and analytic functionals. (English) Zbl 1293.30005

A series \(\sum_{n=0}^\infty a_n\) is said to be summable to a number \(\beta\) by Borel’s method (B\('\)) if the power series \(\sum_{n=0}^\infty a_n t^n/n!\) is convergent for all \(t\) and \[ \int_0^\infty e^{-t} \left(\sum_{n=0}^\infty a_n t^n/n!\right) \, d t=\beta. \] The main theorem shows that a formal power series that is uniformly (B\('\)) summable on a circle of radius \(r>0\) necessarily has radius of convergence at least \(r\). The authors apply the result to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. Also, they use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon.

MSC:

30B10 Power series (including lacunary series) in one complex variable
40G10 Abel, Borel and power series methods
30D15 Special classes of entire functions of one complex variable and growth estimates
46F15 Hyperfunctions, analytic functionals
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References:

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