Interpolating sequences for holomorphic functions of restricted growth
The authors define \(A_\alpha\)-interpolating sequences for the algebra of holomorphic functions of order at most \(\alpha>0\) \[ A_\alpha=\{f\in H(\mathbb{D}):\sup_{z\in\mathbb{D}} (1-| z| )^\alpha\log^+| f(z)| <\infty\} \] in the unit disk \(\mathbb{D}\) of \(\mathbb{C}\) as those sequences \(\{a_k\}_k\subset\mathbb{D}\) such that for every sequence of values \(\{v_k\}\) with \[ \sup\limits_{k\in\mathbb{N}}(1-| a_k| )^\alpha \log^+| v_k| <\infty \] there exists \(f\in A_\alpha\) such that \(f(a_k)=v_k\) for all \(k\in\mathbb{N}\). Main result of the paper is the following. \textbf{Theorem 1.} A sequence \(\{a_k\}_k\subset\mathbb{D}\) is \(A_\alpha\)-interpolating if and only if \[ \sup_{k\in\mathbb{N}} (1-| a_k| )^\alpha N(a_k,1/2)<\infty. \]
scientific article; zbMATH DE number 1865541