Zero Sets for Spaces of Analytic Functions
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, p. 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 : 2017-05-10
Revised : 2017-06-16
Accepted : 2017-11-07
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3210
Classification:  30H20,  30B20,  30C15,  60G15
Keywords: Bergman, Bargmann, Fock, Gaussian, random
@article{AIF_2018__68_6_2311_0,
     author = {Lyons, Russell and Zhai, Alex},
     title = {Zero Sets for Spaces of Analytic Functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     pages = {2311-2328},
     doi = {10.5802/aif.3210},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_6_2311_0}
}
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/item/AIF_2018__68_6_2311_0/

[1] Bomash, Gregory A Blaschke-type product and random zero sets for Bergman spaces, Ark. Mat., Tome 30 (1992) no. 1, pp. 45-60 | Article | MR 1171094 | Zbl 0764.30029

[2] Chistyakov, Gennadiy P.; Yurii I. Lyubarskii; Pastur, Leonid Andreevich On completeness of random exponentials in the Bargmann-Fock space, J. Math. Phys., Tome 42 (2001) no. 8, pp. 3754-3768 | Article | MR 1845217 | Zbl 1009.42005

[3] Duren, Peter; Schuster, Alexander Bergman Spaces, American Mathematical Society, Mathematical Surveys and Monographs, Tome 100 (2004), x+318 pages | Article | MR 2033762 | Zbl 1059.30001

[4] Hedenmalm, Haakan; Korenblum, Boris; Zhu, Kehe Theory of Bergman Spaces, Springer, Graduate Texts in Mathematics, Tome 199 (2000), x+286 pages | Article | MR 1758653 | Zbl 0955.32003

[5] Horowitz, Charles Zeros of functions in the Bergman spaces, Duke Math. J., Tome 41 (1974), pp. 693-710 http://projecteuclid.org/euclid.dmj/1077310733 | MR 0357747 | Zbl 0293.30035

[6] Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint Zeros of Gaussian Analytic Functions and Determinantal Point Processes, American Mathematical Society, University Lecture Series, Tome 51 (2009), x+154 pages | Article | MR 2552864 | Zbl 1190.60038

[7] Kac, Mark On the average number of real roots of a random algebraic equation, Bull. Am. Math. Soc., Tome 49 (1943), pp. 314-320 | Article | MR 0007812 | Zbl 0060.28602

[8] Kac, Mark A correction to “On the average number of real roots of a random algebraic equation”, Bull. Am. Math. Soc., Tome 49 (1943), 938 pages | Article | MR 0009655 | Zbl 0060.28603

[9] Kahane, Jean-Pierre Une inégalité du type de Slepian et Gordon sur les processus gaussiens, Isr. J. Math., Tome 55 (1986) no. 1, p. 109-110 | Article | MR 858463 | Zbl 0611.60034

[10] Leblanc, Emile A probabilistic zero set condition for the Bergman space, Mich. Math. J., Tome 37 (1990) no. 3, pp. 427-438 | Article | MR 1077326 | Zbl 0717.30008

[11] Mateljević, Miodrag; Pavlović, Miroslav L p -behavior of power series with positive coefficients and Hardy spaces, Proc. Am. Math. Soc., Tome 87 (1983) no. 2, pp. 309-316 | Article | MR 681840 | Zbl 0524.30023

[12] Nowak, Maria; Waniurski, Piotr Random zero sets for Bergman spaces, Math. Proc. Camb. Philos. Soc., Tome 134 (2003) no. 2, pp. 337-345 | MR 1972142 | Zbl 1034.30004

[13] Paley, Raymond E. A. C.; Wiener, Norbert Fourier Transforms in the Complex Domain, American Mathematical Society, American Mathematical Society Colloquium Publications, Tome 19 (1934), x+184 pages | Zbl 0011.01601

[14] Peres, Yuval; Virág, Bálint Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Math., Tome 194 (2005) no. 1, pp. 1-35 | Article | MR 2231337 | Zbl 1099.60037

[15] Rice, Stephen O. Mathematical analysis of random noise, Bell System Tech. J., Tome 23 (1944), pp. 282-332 | MR 0010932 | Zbl 0063.06485

[16] Rice, Stephen O. Mathematical analysis of random noise, Bell System Tech. J., Tome 24 (1945), pp. 46-156 | MR 0011918 | Zbl 0063.06487

[17] Sedletskiĭ, Anatoly M. Zeros of analytic functions of the classes A p , Current Problems in Function Theory (Russian) (Teberda, 1985), Rostov. Gos. Univ. (1987), p. 24-29, 177 | MR 1054432

[18] SevastʼYanov, E. A.; Dolgoborodov, A. A. Zeros of functions in weighted spaces with mixed norm, Math. Notes, Tome 94 (2013) no. 1-2, pp. 266-280 (Translation of Mat. Zametki 94 (2013), no. 2, p. 279–294) | Article | MR 3206088 | Zbl 1277.30048

[19] Shapiro, Harold S.; Shields, Allen L. On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z., Tome 80 (1962), pp. 217-229 | MR 0145082 | Zbl 0115.06301

[20] Shapiro, Joel H. Zeros of functions in weighted Bergman spaces, Mich. Math. J., Tome 24 (1977) no. 2, pp. 243-256 http://projecteuclid.org/euclid.mmj/1029001887 | MR 0463454 | Zbl 0376.30009

[21] Shapiro, Joel H. Zeros of random functions in Bergman spaces, Ann. Inst. Fourier, Tome 29 (1979) no. 4, pp. 159-171 | MR 558594 | Zbl 0403.46026

[22] Sodin, Mikhail Zeroes of Gaussian analytic functions, European Congress of Mathematics, European Mathematical Society (2005), pp. 445-458 | MR 2185759 | Zbl 1073.60058

[23] Stokes, George G. Note on the determination of arbitrary constants which appear as multipliers of semi-convergent series, Proc. Camb. Philos. Soc., Tome VI (1889), pp. 362-366 | Zbl 22.0297.01

[24] Zhu, Kehe Zeros of functions in Fock spaces, Complex Variables Theory Appl., Tome 21 (1993) no. 1-2, pp. 87-98 | MR 1276563 | Zbl 0799.30024

[25] Zhu, Kehe Analysis on Fock Spaces, Springer, Graduate Texts in Mathematics, Tome 263 (2012), x+344 pages | Article | MR 2934601 | Zbl 1262.30003