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Random sequences interpolating with probability one. (English) Zbl 0831.30020

For a sequence \(\{r_n\}_{n \geq 0}\) of points in \((0,1)\) the author considers the question of whether the sequence \(\{r_n e^{i \theta_n}\}_{n \geq 0}\) of points of the unit disk is interpolating for \(H^\infty\) for almost all choices of \(\theta_n\). The main result shows that the sequence \(\{r_n e^{i \theta_n}\}\) is interpolating with probability 1 if \(\sum_{k \geq 0} 2^{-k} N_k^2 < \infty\) and is interpolating with probability 0 if \(\sum_{k \geq 0} 2^{-k} N_k^2 = \infty\), where \(N_k\) is the number of terms of \(\{r_n\}_{n \geq 0}\) in \([1 - 2^{- k}, 1 - 2^{- k - 1})\).

MSC:

30D50 Blaschke products, etc. (MSC2000)
30B20 Random power series in one complex variable
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