Статья
2017
On zeros of functions rapidly growing in generalized Bergman spaces
E. A. Sevast’yanov
Русская математика
https://doi.org/10.3103/S1066369X17110068
Abstract / Full Text
The zero-sets of rapidly growing functions which belong to the Bergman spaces and more general spaces of analytic functions with mixed norms have no clear-cut description. A range of exact necessary conditions on the moduli of zeros of such functions presented in the paper show the impossibility to obtain such a description in more or less clear geometrical terms.
Author information
- National Research Nuclear University (MIFI), Kashirskoe sh. 31, Moscow, 115409, RussiaE. A. Sevast’yanov
References
- Li, S. and Stevich, S. “Differentiation of a Composition as an Operator from Spaces with Mixed Norm to Bloch α-spaces”, Sb.Math. 199, No. 11–12, 1847–1857 (2008).
- Horowitz, C. “Zeros of Functions in the Bergman Spaces”, DukeMath. J. 41, No. 4, 693–710 (1974).
- Dzhrbashyan, M. M. “On the Problem of Presentability of Analytic Functions”, Soobshcheniya Instituta Matematiki iMekhaniki AN Arm. SSR, No. 2, 3–40 (1948) [in Russian].
- Korenblum, B. “An Extension of the Nevanlinna Theory”, ActaMath. 135, No. 3–4, 187–219 (1975).
- Hedenmalm, H., Korenblum, B. and Zhu, K.Theory of Bergman Spaces. Graduate Texts in Mathematics (Springer-Verlag, New York, 2000), Vol. 199.
- Seip, K. “On Korenblum‘s Density Condition for the Zero Sequences of A −α”, J. Anal. Math. 67, No. 1, 307–322 (1995).
- Luecking, D. H. “Zero Sequences for Bergman Spaces”, Complex Variables Theory Appl. 30, No. 4, 345–362 (1996).
- Blasco, O., Kukuryka, A. and Nowak, M. “Luecking’s Condition for Zeros of Analytic Functions”, Ann. Univ.Mariae Curie-Sklodowska Sect. A 58, 1–15 (2004).
- Seip, K. “An Extension of the Blaschke Condition”, J. LondonMath. Soc. (2) 51, No. 3, 545–558 (1995).
- Bruna, J. and Massaneda, X. “Zero Sets of Holomorphic Functions in the Unit Ball with Slow Growth”, J. Anal.Math. 66, No. 1, 217–252 (1995).
- Kudasheva, E. G. and Khabibullin, B. N. “Distribution of Zeros of Holomorphic Functions of Moderate Growth in the Unit Disk and the Representation ofMeromorphic Functions in it”, Sb.Math. 200, No. 9–10, 1353–1382 (2009).
- Dolgoborodov, A. A. and Sevast’yanov, E. A. “Extremal Properties of Sequences of Zeros of Analytic Functions from Spaces withMixed Norm”, Sb.Math. 202, No. 7–8, 1085–1103 (2011).
- Sevast’yanov, E. A. and Dolgoborodov, A. A. “Zeros of Functions in Weighted Spaces with Mixed Norm”, Math. Notes 94, No. 1–2, 266–280 (2013).
- Biryukov, L. N. “Distribution of Zeros of Functions from the Classes Ap,−1,l”, MoscowUniv.Math. Bull. 60, No. 5, 12–18 (2005).
- Sevast’yanov, E. A. and Dolgoborodov, A. A. “On the Distribution of Zeros of Functions from Weighted Bergman Spaces”, Math.Notes 86, No. 1–2, 93–106 (2009).