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Zero sets and random zero sets in certain function spaces. (English) Zbl 0819.30003

The authors discuss primarily the zero sets of the space \(B^ p\), \(p > 1\), defined as the collection of functions \(f\) analytic in the unit disk \(U\) such that \[ \int_ U \biggl (\log^ + \bigl | f(z) \bigr | \biggr)^ p dA(z) < + \infty. \] Here \(dA(z)\) denotes Lebesgue measure on \(\mathbb{C}\). A. Heilper has shown [Isr. J. Math. 34, 1-11 (1979; Zbl 0498.30040)] that the zero sets of \(B^ 1\) are completely characterized by the “Blaschke-type” condition \[ \sum_ j \bigl( 1 - | z_ j | \bigr)^ 2 < + \infty. \] In the present paper, it is shown that when \(p > 0\), a characterization solely in terms of the moduli of the zeros is impossible; this is done by obtaining necessary conditions for the zeros on a single radius, which are stronger than certain best possible conditions due to E. Beller [Israel J. Math. 22, 68-80 (1975; Zbl 0322.30028)]. However, it is proved that a condition very close to that of Beller is “almost surely” sufficient. The authors adapt here a probabilistic approach, introduced by E. LeBlanc [Mich. Math. J. 37, No. 3, 427-438 (1990; Zbl 0717.30008)] for studying the zero sets of Bergman spaces. Finally, using an idea of the reviewer, the authors strengthen some of the conditions on Bergman space zero sets obtained in a recent paper of Ch. Horowitz [J. Anal. Math. 62, 323- 348 (1994; Zbl 0795.30006)].
Reviewer: K.Seip

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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