Random Blaschke products

Cochran, W. George

doi:10.1090/S0002-9947-1990-1022163-8

Random Blaschke products
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by W. George Cochran PDF

Trans. Amer. Math. Soc. 322 (1990), 731-755 Request permission

Abstract:

Let $\{ {\theta _n}(\omega )\}$ be a sequence of independent random variables uniformly distributed on $[0,2\pi ]$, and let ${z_n}(\omega ) = {r_n}{e^{i{\theta _n}(\omega )}}$ for a fixed but arbitrary sequence of radii ${r_n}$ satisfying the Blaschke condition $\sum {(1 - {r_n}) < \infty }$. We show that the random Blaschke product with zeros ${z_n}(\omega )$ is almost surely not in the little Bloch space, and we describe necessary conditions and sufficient conditions on the radii ${r_n}$ so that $\{ {z_n}(\omega )\}$ is almost surely an interpolating sequence.

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