Radial limits of interpolating Blaschke products
Let \(B\) be a Blaschke product with the zero sequence \((z_n)_n\subset\mathbb D=\) the unit disc. We say that \(B\) is an interpolating Blaschke product if for any bounded sequence \((w_n)_n\subset\mathbb C\) there exists a bounded holomorphic function \(f\) in \(\mathbb D\) with \(f(z_n)=w_n\) for all \(n\). The authors investigate the following interpolation problem. Given points \((\lambda_n)_n\subset\partial\mathbb D\), \((a_n)_n\subset\overline{\mathbb D}\) find conditions under which there exists an interpolating Blaschke product \(B\) such that for each \(n\) the radial limit of \(B\) at \(\lambda_n\) equals \(a_n\). The main result of the paper says that if \((\lambda_n)_n\subset \partial\mathbb D\), then the problem has a solution iff \(\inf_n| a_n| >0\). See also: [\textit{G. Cargo}, J. Math. Anal. Appl. 71, 287-296 (1979; Zbl 0428.30021); \textit{C. Belna, F. Carroll, G. Piranian}, Trans. Am. Math. Soc. 280, 695--702 (1983; Zbl 0549.30023); \textit{C. Belna, P. Colwell, G. Piranian}, Proc. Am. Math. Soc. 93, 267--271 (1985; Zbl 0582.30022)].
scientific article; zbMATH DE number 2156227
scientific article