Values of BMOA functions of interpolating sequences
Let \(\rho (z,w)=| (z-w)/(1-\bar zw)|\) denote the pseudo- hyperbolic distance between z, \(w\in {\mathbb{D}}\) and let \(\{z_ n\}\) be an interpolating sequence in \({\mathbb{D}}\), i.e. a sequence satisfying \(\prod_{m\neq n}\rho (z_ m,z_ n)\geq \rho >0\) for all n. The purpose of this interesting paper is to give a complete characterization of those sequences of complex numbers \(a_ n\) for which there exists a function \(f\in BMOA\), the space of analytic functions of bounded mean oscillation, satisfying \(f(z_ n)=a_ n\) for all n. This extends former results of \textit{L. Carleson} [Am. J. Math. 80, 921-930 (1958; Zbl 0085.065)] for \(H^{\infty}\), of \textit{H. S. Shapiro} and \textit{A. L. Shields} [Am. J. Math. 83, 513-532 (1961; Zbl 0112.297)] for \(H^ p\) \((1\leq p<\infty)\) and of \textit{V. Kabajla} [Litov. Mat. Sb. 3, No.1, 141-147 (1963; Zbl 0171.037)] for \(H^ p\) \((0<p<1)\).
scientific article; zbMATH DE number 3946553