Zero Sets for Spaces of Analytic Functions
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2311-2328.

We show that under mild conditions, a Gaussian analytic function F that a.s. does not belong to a given weighted Bergman space or Bargmann–Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro [21] on Bergman spaces and allows us to resolve a question of Zhu [24] on Bargmann–Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro [21] on Bergman spaces and allowing us to strengthen a result of Zhu [24] on Bargmann–Fock spaces.

On montre que sous des conditions faibles, une fonction analytique gaussienne F qui n’appartient pas p.s. à un espace pondéré de Bergman ou de Bargmann–Fock donné a p.s. la propriété qu’il n’existe pas de fonction non-nulle dans cette espace qui s’annule où F s’annule. Ceci démontre une conjecture de Shapiro [21] sur les espaces de Bergman et nous permet de résoudre une question de Zhu [24] sur les espaces de Bargmann–Fock. On donne aussi un résultat similaire sur la réunion de deux (ou plus) tels ensembles de zéros, montrant ainsi une autre conjecture de Shapiro [21] sur les espaces de Bergman et nous permettant de renforcer un résultat de Zhu [24] sur les espaces de Bargmann–Fock.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3210
Classification: 30H20,  30B20,  30C15,  60G15
Keywords: Bergman, Bargmann, Fock, Gaussian, random
Lyons, Russell 1; Zhai, Alex 2

1 Department of Mathematics 831 E. 3rd St. Indiana University Bloomington, IN 47405-7106 (USA)
2 Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305 (USA)
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Lyons, Russell; Zhai, Alex. Zero Sets for Spaces of Analytic Functions. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2311-2328. doi : 10.5802/aif.3210. https://aif.centre-mersenne.org/articles/10.5802/aif.3210/

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