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St. Petersburg Mathematical Journal

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Balayage of measures and subharmonic functions to a system of rays. I. The classical case


Authors: B. N. Khabibullin and A. V. Shmeleva
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 1.
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
Published electronically: December 3, 2019
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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.


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Additional 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

DOI: https://doi.org/10.1090/spmj/1589
Keywords: Entire function, sequence of zeros, subharmonic function, Riesz measure, balayage
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
Article copyright: © Copyright 2019 American Mathematical Society