Zeros of random functions in Bergman spaces

Shapiro, Joel H.

doi:10.5802/aif.772

Shapiro, Joel H.

Annales de l'Institut Fourier, Volume 29 (1979) no. 4, pp. 159-171.

Abstract
Résumé

Suppose $μ$ is a finite positive rotation invariant Borel measure on the open unit disc $Δ$ , and that the unit circle lies in the closed support of $μ$ . For $0 < p < \infty$ the Bergman space $A_{μ}^{p}$ is the collection of functions in $L^{p} (μ)$ holomorphic on $Δ$ . We show that whenever a Gaussian power series $f (z) = Σ ζ_{n} a_{n} z^{n}$ almost surely lies in $A_{μ}^{p}$ but not in $⋃_{q > p} A_{μ}^{p}$ , then almost surely: a) the zero set $Z (f)$ of $f$ is not contained in any $A_{μ}^{q}$ zero set ( $q > p$ , and b) $Z (f + 1) \cup Z (f - 1)$ is not contained in any $A_{μ}^{q}$ zero set.

Soit $μ$ une mesure positive invariante par rotation sur le disque unité ouvert $Δ$ , telle que le support de $μ$ contienne le cercle unité. Pour $0 < p < \infty$ l’ensemble des fonctions de $L^{p} (μ)$ qui sont holomorphes sur $Δ$ s’appelle l’espace de Bergman $A_{μ}^{p}$ . Nous montrons que, lorsque la série de puissances $f (z) = Σ ζ_{n} a_{n} z^{n}$ à coefficients gaussiens indépendants est presque sûrement dans $A_{μ}^{p} ∖ ⋃_{q > p} A_{μ}^{p}$ , alors il est presque sûr que : a) $Z (f)$ , ensemble des zéros de $f$ , n’est contenu dans aucun ensemble $Z A_{μ}^{q}$ (c’est-à-dire $Z (g)$ , $g \in A_{μ}^{q} ∖ {0}$ , $q > p$ ), et b) $Z (f + 1) \cup Z (f - 1)$ n’est contenu dans aucun $Z A_{μ}^{q}$ .

MR Zbl

DOI: 10.5802/aif.772

@article{AIF_1979__29_4_159_0,
     author = {Shapiro, Joel H.},
     title = {Zeros of random functions in {Bergman} spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {159--171},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {4},
     year = {1979},
     doi = {10.5802/aif.772},
     mrnumber = {81h:30054},
     zbl = {0403.46026},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.772/}
}

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Shapiro, Joel H. Zeros of random functions in Bergman spaces. Annales de l'Institut Fourier, Volume 29 (1979) no. 4, pp. 159-171. doi : 10.5802/aif.772. http://archive.numdam.org/articles/10.5802/aif.772/

References
Cited by

[1] X. Fernique, Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci., Paris, 270 (1970), 1698-1699. | MR | Zbl

[2] Ch. Horowitz, Zeros of functions in Bergman spaces, Duke Math. J., 41 (1974), 693-710. | MR | Zbl

[3] J.P. Kahane, Some Random Series of Functions, D.C. Heath and Co., Lexington, MA, 1968. | MR | Zbl

[4] W. Rudin, Zeros of holomorphic functions in balls, Indag. Math., 38 (1976), 57-65. | MR | Zbl

[5] W. Rudin, Principles of Real Analysis, 3rd ed., McGraw-Hill, New York, 1976. | Zbl

[6] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974. | MR | Zbl

[7] J. H. Shapiro, Zeros of functions in weighted Bergman spaces, Michigan Math. J., 24 (1977), 243-256. | MR | Zbl

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