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Random sequences with prescribed radii in the unit ball. (English) Zbl 0865.41007

Given a sequence \(\{a_k \}_k\) in the unit ball of \(\mathbb{C}^n\) consider the sequence \(\{a_k (\omega) \}_k\) obtained by applying to each \(a_k\) a random rotation. We give conditions on \(\{|a_k |\}_k\) so that the zero-varieties whose irreducible components are hyperplanes \(X_{a_k (\omega)}= \{z\in \mathbb{B}^n: z\cdot \overline {a}_k (\omega)= |a_k |^2\}\) are contained in zero sets of Hardy and Bergman functions. Also, we give sharp conditions for \(\{a_k (\omega) \}_k\) to be interpolating for weighted Bergman spaces almost surely, and for \(\{a_k (\omega) \}_k\) to be a finite union of separated sequences (in terms of the Gleason distance). Some of these results extend their previous corresponding versions in the disc.

MSC:

41A05 Interpolation in approximation theory
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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