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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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[1] Beller, E., Zeros of A^P functions and related classes of analytic functions, Israel J. Math., 22, 68-80 (1975) · Zbl 0322.30028 · doi:10.1007/BF02757275
[2] Bomash, G., A Blaschke-type product and random zero sets for Bergman spaces, Arkiv für Math., 30, 45-60 (1992) · Zbl 0764.30029 · doi:10.1007/BF02384861
[3] Hayman, W. K.; Korenblum, B., An extension of the Riesz-Herglotz formula, Ann. Acad. Sci. Fenn. A.I. Math., 2, 175-201 (1976) · Zbl 0416.30019
[4] Hayman, W. K.; Korenblum, B., A critical growth rate of functions regular in a disk, Mich. Math. J., 27, 21-30 (1980) · Zbl 0417.30023 · doi:10.1307/mmj/1029002305
[5] Heilper, A., The zeros of functions in Nevanlinna’s area class, Israel J. Math., 34, 1-11 (1979) · Zbl 0498.30040 · doi:10.1007/BF02761820
[6] C. Horowitz,Zeros of Functions in the Bergman Spaces, Ph.D. Thesis, Univ. of Michigan, 1974. · Zbl 0293.30035
[7] Horowitz, C., Zeros of functions in the Bergman spaces, Duke Math. J., 41, 693-710 (1974) · Zbl 0293.30035 · doi:10.1215/S0012-7094-74-04175-1
[8] Horowitz, C., Some conditions on Bergman space zero sets, J. Analyse Math., 62, 323-348 (1994) · Zbl 0795.30006 · doi:10.1007/BF02835961
[9] LeBlanc, E., A probabilistic zero set condition for the Bergman space, Mich. Math. J., 37, 427-438 (1990) · Zbl 0717.30008 · doi:10.1307/mmj/1029004200
[10] K. Seip,Beurling type density theorems in the unit disk, Inventiones Math. (to appear). · Zbl 0789.30025
[11] Shapiro, H. S.; Shields, A. L., On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z., 80, 217-229 (1962) · Zbl 0115.06301 · doi:10.1007/BF01162379
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