Random Blaschke products. (English) Zbl 0709.30031

The paper deals with infinite Blaschke products whose zeros are chosen in a random way. The motivation for the author to study such products are twofold. Firstly, the question asked by D. Sarason: will an infinite Blaschke product whose zeros are chosen randomly (in some sense) be in the little Bloch space? Secondly, the following old result due to A. G. Naftalevich: given a sequence \(\{r_ n\}\), \(0<r_ n<1\), satisfying the Blaschke condition \(\sum (1-r_ n)<\infty\), one can find an interpolating sequence of complex numbers \(z_ n\) with \(| z_ n| =r_ n\) for every n. It turns out that Theorem 1 of the paper answers negatively the above question of Sarason.
Theorem 1. Let \(\{r_ n\}\) be a sequence of numbers such that \(\sum (1- r_ n)<\infty\) and let \(\{\theta_ n(\omega)\}\) be a sequence of independent random variables which are uniformly distributed on [0,2\(\pi\) ]. Then for the Blaschke product B with zeros \(r_ ne^{i\theta_ n(\omega)}\) one has \[ \limsup_{n\to \infty}(1-| z_ n|)| B'(z_ n)| =1\quad almost\quad surely. \] The paper also contains a detailed proof of the above mentioned result of A. G. Naftalevich. The methods used in the paper form a combination of analytic and probabilistic tools.
Reviewer: J.Janas


30D50 Blaschke products, etc. (MSC2000)
30B20 Random power series in one complex variable
46E15 Banach spaces of continuous, differentiable or analytic functions
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