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Thin interpolating sequences in the disk. (English) Zbl 1179.30058

The author gives a new proof of an important result of C. Sundberg and 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 K. Dyakonov and A. Nicolau [Trans. Am. Math. Soc. 359, No. 9, 4449–4465 (2007; Zbl 1141.30010)].

MSC:

30J10 Blaschke products
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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