Efficient analytic continuation of power series with vector-valued coefficients
The main result of the paper is the following continuation theorem. Let \(X\) be a complex Banach space and let \[ f(z)= \sum^\infty_{n=0} f_n/z^{n+1},\quad f_n\in X,\quad n\geq 0. \] Assume that \(E\) is a logarithmically convex compact subset of \(\mathbb{C}\) such that \(\mathbb{C}\backslash E\) is connected and \(0\in E\). Let \(L\) be a logarithmic diagram of \(E\) of direction \(\beta\). Then \(f\in {\mathcal O}(\widehat{\mathbb{C}}\backslash E,X)\) iff there exists \(\varphi\in{\mathcal O}(\mathbb{C},X)\) such that: the order of \(\varphi\) is \(\leq 1\), \(\varphi\) is of exponential type in \(\{re^{i\theta}: r\geq 0,|\theta+\beta|<\pi/2\}\), \(\varphi(n)=f_n\) for \(n\geq 0\), and \[ \limsup_{r\to+\infty} (1/r)\log|\varphi(re^{i\theta})|:= h_\varphi(\theta)\leq K_L(-\theta):= \sup\{\text{Re}(\zeta e^{i\theta}):\zeta\in L\} \] for \(|\theta+\beta|< \pi/2\).
scientific article; zbMATH DE number 920932