Skip to main content
SpringerLink
Log in

On zeros of functions rapidly growing in generalized Bergman spaces

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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).

    Article  MathSciNet  Google Scholar 

  2. Horowitz, C. “Zeros of Functions in the Bergman Spaces”, DukeMath. J. 41, No. 4, 693–710 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  3. 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].

    Google Scholar 

  4. Korenblum, B. “An Extension of the Nevanlinna Theory”, ActaMath. 135, No. 3–4, 187–219 (1975).

    Google Scholar 

  5. Hedenmalm, H., Korenblum, B. and Zhu, K.Theory of Bergman Spaces. Graduate Texts in Mathematics (Springer-Verlag, New York, 2000), Vol. 199.

  6. Seip, K. “On Korenblum‘s Density Condition for the Zero Sequences of A −α”, J. Anal. Math. 67, No. 1, 307–322 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  7. Luecking, D. H. “Zero Sequences for Bergman Spaces”, Complex Variables Theory Appl. 30, No. 4, 345–362 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  8. 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).

    MATH  MathSciNet  Google Scholar 

  9. Seip, K. “An Extension of the Blaschke Condition”, J. LondonMath. Soc. (2) 51, No. 3, 545–558 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  10. 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).

    Article  MATH  MathSciNet  Google Scholar 

  11. 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).

    Article  MATH  MathSciNet  Google Scholar 

  12. 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).

    Article  MATH  MathSciNet  Google Scholar 

  13. 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).

    Article  MATH  MathSciNet  Google Scholar 

  14. Biryukov, L. N. “Distribution of Zeros of Functions from the Classes Ap,−1,l”, MoscowUniv.Math. Bull. 60, No. 5, 12–18 (2005).

    Google Scholar 

  15. 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).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research Nuclear University (MIFI), Kashirskoe sh. 31, Moscow, 115409, Russia

    E. A. Sevast’yanov

Authors
  1. E. A. Sevast’yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. A. Sevast’yanov.

Additional information

Original Russian Text © E.A. Sevast’yanov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 11, pp. 46–59.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sevast’yanov, E.A. On zeros of functions rapidly growing in generalized Bergman spaces. Russ Math. 61, 40–52 (2017). https://doi.org/10.3103/S1066369X17110068

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X17110068

Keywords

Search

Navigation