Zero Sets for Spaces of Analytic Functions

Lyons, Russell; Zhai, Alex

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2311-2328.

On montre que sous des conditions faibles, une fonction analytique gaussienne $\mathbf{F}$ 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 $\mathbf{F}$ 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.

We show that under mild conditions, a Gaussian analytic function $\mathbf{F}$ 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 $\mathbf{F}$ 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.

Classification : 30H20, 30B20, 30C15, 60G15
Keywords: Bergman, Bargmann, Fock, Gaussian, random
Mot clés : Bergman, Bargmann, Fock, gaussienne, aléatoire

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

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

[3] Duren, Peter; Schuster, Alexander Bergman Spaces, Mathematical Surveys and Monographs, 100, American Mathematical Society, 2004, x+318 pages | DOI | MR | Zbl

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

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

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

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

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

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

[10] LeBlanc, Emile A probabilistic zero set condition for the Bergman space, Mich. Math. J., Volume 37 (1990) no. 3, pp. 427-438 | DOI | MR | Zbl

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

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

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

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

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

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

[17] Sedletskiĭ, Anatoly M. Zeros of analytic functions of the classes $A_{\infty }^p$, Current Problems in Function Theory (Russian) (Teberda, 1985), Rostov. Gos. Univ., 1987, p. 24-29, 177 | MR

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

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

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

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

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

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

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

[25] Zhu, Kehe Analysis on Fock Spaces, Graduate Texts in Mathematics, 263, Springer, 2012, x+344 pages | DOI | MR | Zbl