Skip to Main Content

St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Balayage of measures and subharmonic functions to a system of rays. I. The classical case
HTML articles powered by AMS MathViewer

by B. N. Khabibullin and A. V. Shmeleva
Translated by: S. V. Kislyakov
St. Petersburg Math. J. 31 (2020), 117-156
DOI: https://doi.org/10.1090/spmj/1589
Published electronically: December 3, 2019
PDF | Request permission

Abstract:

Classical balayage of measures and subharmonic functions to a system $S$ of rays with common origin on the complex plane $\mathbb {C}$ is developed. For a subharmonic function $v$ on $\mathbb {C}$, this makes it possible to construct another subharmonic function on $\mathbb {C}$ that is harmonic outside $S$ and coincides with $v$ on $S$. This is applied to the study of the relationship between the growth of an entire function on $S$ and the distributions of its zeros, the investigation of conditions for completely regular growth of entire and subharmonic functions, the analysis of noncompleteness for systems of exponentials in spaces of holomorphic functions in nonconvex unbounded open sets that narrow down near infinity. The present text is the first part of the paper, and it contains also the necessary preliminaries for the construction of a new type of balayage of the second kind on $S$, which will be discussed in the second part.
References
  • H. Poincare, Sur les Equations aux Derivees Partielles de la Physique Mathematique, Amer. J. Math. 12 (1890), no. 3, 211–294 (French). MR 1505534, DOI 10.2307/2369620
  • H. Poincaré, Théorie du potentiel Newtonien, Paris, 1899.
  • Ch.-J. de La Vallée-Poussin, Le potentiel logarithmique, balayage et représentation conforme, Löwen, 1949.
  • N. S. Landkof, Osnovy sovremennoĭ teorii potentsiala, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0214795
  • J. Bliedtner and W. Hansen, Potential theory, Universitext, Springer-Verlag, Berlin, 1986. An analytic and probabilistic approach to balayage. MR 850715, DOI 10.1007/978-3-642-71131-2
  • B. N. Khabibullin, Dual representation of superlinear functionals and its applications in the theory of functions. I, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 4, 205–224 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 4, 835–852. MR 1857717, DOI 10.1070/IM2001v065n04ABEH000354
  • B. N. Khabibullin, Dual representation of superlinear functionals and its applications in the theory of functions. II, Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), no. 5, 167–190 (Russian, with Russian summary); English transl., Izv. Math. 65 (2001), no. 5, 1017–1039. MR 1874358, DOI 10.1070/IM2001v065n05ABEH000361
  • B. N. Khabibullin, Smallness of growth on the imaginary axis of entire functions of exponential type with given zeros, Mat. Zametki 43 (1988), no. 5, 644–650, 702 (Russian); English transl., Math. Notes 43 (1988), no. 5-6, 372–375. MR 954347, DOI 10.1007/BF01158843
  • B. N. Khabibullin, Sets of uniqueness in spaces of entire functions of one variable, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 1101–1123 (Russian); English transl., Math. USSR-Izv. 39 (1992), no. 2, 1063–1084. MR 1149889, DOI 10.1070/IM1992v039n02ABEH002237
  • B. N. Khabibullin, On the growth of entire functions of exponential type along the imaginary axis, Mat. Sb. 180 (1989), no. 5, 706–719 (Russian); English transl., Math. USSR-Sb. 67 (1990), no. 1, 149–163. MR 1007471, DOI 10.1070/SM1990v067n01ABEH003735
  • B. N. Khabibullin, On the growth of entire functions of exponential type with given zeros along a line, Anal. Math. 17 (1991), no. 3, 239–256 (Russian, with English summary). MR 1198963, DOI 10.1007/BF01907345
  • B. N. Khabibullin, Helly’s theorem and shifts of sets. II. Support function, exponential systems, entire functions, Ufa Math. J. 6 (2014), no. 4, 122–134. MR 3427562, DOI 10.13108/2014-6-4-122
  • B. N. Khabibullin, On the growth of entire functions of exponential type with zeros near a straight line, Mat. Zametki 70 (2001), no. 4, 621–635 (Russian, with Russian summary); English transl., Math. Notes 70 (2001), no. 3-4, 560–573. MR 1884199, DOI 10.1023/A:1012393106283
  • B. N. Khabibullin, Helly’s theorem and shifts of sets. II. Support function, exponential systems, entire functions, Ufa Math. J. 6 (2014), no. 4, 122–134. MR 3427562, DOI 10.13108/2014-6-4-122
  • Paul Malliavin and L. A. Rubel, On small entire functions of exponential type with given zeros, Bull. Soc. Math. France 89 (1961), 175–206. MR 141778
  • Lee A. Rubel, Entire and meromorphic functions, Universitext, Springer-Verlag, New York, 1996. With the assistance of James E. Colliander. MR 1383095, DOI 10.1007/978-1-4612-0735-1
  • B. Ya. Levin, Distribution of zeros of entire functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0087740
  • B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006, DOI 10.1090/mmono/150
  • A. A. Gol′dberg, B. Ya. Levin, and I. V. Ostrovskiĭ, Entire and meromorphic functions, Complex analysis. One variable, 1 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1991, pp. 5–185, 256 (Russian). MR 1155417
  • Anatoly A. Goldberg and Iossif V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. Translated from the 1970 Russian original by Mikhail Ostrovskii; With an appendix by Alexandre Eremenko and James K. Langley. MR 2435270, DOI 10.1090/mmono/236
  • Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
  • Vladimir Azarin, Growth theory of subharmonic functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. MR 2463743
  • Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301332
  • John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
  • John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original. MR 2450237
  • Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
  • N. Bourbaki, Eléments de mathématique. XIII. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre I: Inégalités de convexité. Chapitre II: Espaces de Riesz. Chapitre III: Mesures sur les espaces localement compacts. Chapitre IV: Prolongement d’une mesure; espaces $L^p$, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann & Cie, Paris, 1952 (French). MR 0054691
  • Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
  • B. N. Khabibullin, Zero sequences of holomorphic functions, representation of meromorphic functions, and harmonic minorants, Mat. Sb. 198 (2007), no. 2, 121–160 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 1-2, 261–298. MR 2355445, DOI 10.1070/SM2007v198n02ABEH003837
  • V. Matsaev, I. Ostrovskii, and M. Sodin, Variations on the theme of Marcinkiewicz’ inequality, J. Anal. Math. 86 (2002), 289–317. MR 1894486, DOI 10.1007/BF02786653
  • I. P. Natanson, Teoriya funtsiĭ veshchestvennoĭ peremennoĭ, 3rd ed., Izdat. “Nauka”, Moscow, 1974 (Russian). MR 0354979
  • B. N. Khabibullin and T. Yu. Baĭguskarov, The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function, Mat. Zametki 99 (2016), no. 4, 588–602 (Russian, with Russian summary); English transl., Math. Notes 99 (2016), no. 3-4, 576–589. MR 3507423, DOI 10.4213/mzm10953
  • V. Matsaev and M. Sodin, Distribution of Hilbert transforms of measures, Geom. Funct. Anal. 10 (2000), no. 1, 160–184. MR 1748919, DOI 10.1007/s000390050005
  • T. Yu. Baĭguskarov, G. R. Talipova, and B. N. Khabibullin, Subsequences of zeros for classes of entire functions of exponential type, allocated by restrictions on their growth, Algebra i Analiz 28 (2016), no. 2, 1–33 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 28 (2017), no. 2, 127–151. MR 3593001, DOI 10.1090/spmj/1442
Similar Articles
Bibliographic Information
  • B. N. Khabibullin
  • Affiliation: Department of Mathematics and Informational Technologies, Bashkir State University, ul. Zaki Validi 32, Ufa 450074, Bashkortostan, Russia
  • Email: Khabib-Bulat@mail.ru
  • A. V. Shmeleva
  • Affiliation: Department of Mathematics and Informational Technologies, Bashkir State University, ul. Zaki Validi 32, Ufa 450074, Bashkortostan, Russia
  • Received by editor(s): January 10, 2017
  • Published electronically: December 3, 2019
  • Additional Notes: This work was supported by the Russian Foundation for Basic Research, grant No. 16-01-00024a
  • © Copyright 2019 American Mathematical Society
  • Journal: St. Petersburg Math. J. 31 (2020), 117-156
  • MSC (2010): Primary 30D15; Secondary 30D35, 41A30, 31A05
  • DOI: https://doi.org/10.1090/spmj/1589
  • MathSciNet review: 3932822