Interpolation by Bloch functions
Let \(\Gamma = \{z_ n\}\) be an infinite sequence of distinct points in the unit disk \(\mathbb D = \{z: | z| < 1\}\) having no accumulation points in \(\mathbb D.\) To this sequence, the space \(\ell^\infty(\Gamma),\) which consists of all sequences \(\{a_ n\}\) having the property that \(\sup_ n| a_ n|/\log1/(1-| z_ n|^2)<\infty,\) is associated. Given \(\Gamma,\) consider the linear operator \(T,\) which maps an analytic function \(f\) in \(\mathbb D\) to the sequence \(\{f(z_ n)\}_ n.\) A sequence \(\Gamma\) is said to be a set of interpolation for the Bloch space \(\mathbb B\) if \(T\) maps \(\mathbb B\) onto \(\ell^\infty(\Gamma).\) The aim of this paper is to find a complete geometric characterization of these sets. The author obtains some necessary and sufficient conditions for \(\Gamma\) to be a set of interpolation, as well as discovers some important properties of these sets. The sampling problem, in some sense the dual problem to interpolation, is examined. The natural question about sets of sampling concerns their existence. It turns out that there are no sets of sampling for the Bloch space, and therefore none for the little Bloch space either.
scientific article; zbMATH DE number 1361052