Radius of analyticity of analytic functions on Banach spaces
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.
scientific article; zbMATH DE number 6864718
scientific article