Zeros of random functions in Bergman spaces
Annales de l'Institut Fourier, Volume 29 (1979) no. 4, p. 159-171

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< 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)=Σζ 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)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< 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)=Σζ 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 ZA μ q (c’est-à-dire Z(g), gA μ q {0}, q>p), et b) Z(f+1)Z(f-1) n’est contenu dans aucun ZA μ q .

@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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {29},
     number = {4},
     year = {1979},
     pages = {159-171},
     doi = {10.5802/aif.772},
     zbl = {0403.46026},
     mrnumber = {81h:30054},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1979__29_4_159_0}
}
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://www.numdam.org/item/AIF_1979__29_4_159_0/

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