Radius of analyticity of analytic functions on Banach spaces
type: Item
instance of: scholarly article
title: Radius of analyticity of analytic functions on Banach spaces
zbMATH Open document ID: 1400.46038
author: Christopher Boyd
author: R. A. Ryan
author: Nina Snigireva
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-04-25T00:00:00Z
review text: The primary interest here is in two measures of convergence of power series in real Banach spaces \(E.\) Consider a point \(a \in E\) and a power series \(\sum_{j=0}^\infty P_j(x-a),\) where each \(P_j\) is a continuous \(j\)-homogeneous polynomial on \(E\). The radius of uniform convergence of this series is defined to be \(\sup\{r \;| \;\sum_{j=0}^\infty P_j(x-a)\) converges uniformly in \(B_r(a) \}.\) Next, following \textit{T. Nguyen} [Stud. Math. 191, No. 2, 171--179 (2009; Zbl 1193.32001)], the authors consider the radius of analyticity of a function \(f:B_R(a) \to F\), where \(F\) is another real Banach space. A real analytic function \(f:B_R(a) \to F\) is called fully analytic if for all \(b \in B_R(a)\), the series \(\sum_{j=0}^\infty \frac{1}{j!}d^jf(b)\) converges uniformly to \(f\) in \(B_{[R-||b-a||]}(b)\). (Almost) finally, the radius of analyticity of \(f\) at \(a \in E\) is the largest \(\rho\) such that \(f\) is fully analytic in \(B_\rho(a)\). Lastly, the constant of analyticity \(\mathcal A(E)\) of \(E\) is equal to \(\sup \{\rho > 0 : \text{every power series at }0 \in E \text{ with unit radius of uniform convergence has radius of analyticity}\geq \rho\}\). It is not difficult that for all \(E\), \(\frac{1}{\sqrt 2} \leq \mathcal A(E) \leq 1\). Also, Nguyen [loc. cit.] showed that \(\mathcal A(E) = 1\) if \(E\) is a Hilbert space. It is interesting to note that for real analytic functions the question of equality of the radius of convergence and the radius of analyticity remains open. Here, the authors show that, if there is equality for \(E = \ell_1\), then one has equality for all \(E\). In addition, the authors investigate this and related problems when \(E\) is a real Banach lattice.
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46T25
zbMATH DE Number: 6864718
zbMATH Keywords: real analytic function
zbMATH Keywords: radius of convergence
zbMATH Keywords: Banach lattice
zbMATH Keywords: regular polynomial
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.03.004
OpenAlex ID: W2790631522
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Wikidata QID: Q130173486
DOI: 10.1016/J.JMAA.2018.03.004
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description: scientific article; zbMATH DE number 6864718