Thin interpolating sequences in the disk
The author gives a new proof of an important result of \textit{C. Sundberg} and \textit{T. H. Wolff} [Trans. Am. Math. Soc. 276, 551--581 (1983; Zbl 0536.30025)]: Let \((z_n)\) be a thin interpolating sequence in \(\mathbb{D}=\{|z|<1\}\). Then there are numbers \(\tau_j\in]0,1[\) and \(\gamma_j\in]0,1[\) with \(\lim\tau_j=\lim \lambda_j=1\) such that, for every sequence \((\xi_j)^n_1\) in \(\mathbb{D}\) with \(|(z_j-\xi_j)/(1-\overline z_j\xi_j)|<\tau_j\), then \(\prod_{k:k\neq j}|(\xi_j-\xi_k)/(1-\overline \xi_j\xi_k)|\geq\gamma_j\). He also proves that every asymptotic interpolation problem \(|f(a_n)-w_n|\to 0\), \(\sup_n|w_n|\leq 1\), can be solved by a thin Blaschke product whenever \((a_n)\) is thin. This result appeared implicitly earlier in work of \textit{K. Dyakonov} and \textit{A. Nicolau} [Trans. Am. Math. Soc. 359, No. 9, 4449--4465 (2007; Zbl 1141.30010)].
scientific article; zbMATH DE number 5565264
scientific article