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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2019, Volume 53, Issue 2, Pages 42–58 (Mi faa3597)

On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems

B. N. Khabibullin, F. B. Khabibullin

Bashkir State University, Ufa

Abstract: Let $M$ be a subharmonic function on a domain $D\subset \mathbb C^n$ with Riesz measure $\nu_M$, ${\mathsf Z} \subset D$. As was shown in the first of the preceding articles, if there exists a holomorphic function $f\neq 0$ on $D$, $f ({\mathsf Z}) = 0$, $|f|\leq \exp M$ on $D$, then there is some scale of integral uniform estimates from above of the distribution of the set $\mathsf Z$ in terms of $\nu_M$. In this article we show that for $n = 1$ this result is “almost invertible”. From such scale estimates of the distribution of points of the sequence ${\mathsf Z}:= \{{\mathsf z} _k \}_{k = 1,2, … \subset D \subset \mathbb C$ by $\nu_M$ it follows that there exists a nonzero holomorphic function $f$ in $D$, $f (\mathsf Z) =0$, $|f| \leq \exp M^{\uparrow}$ on $D$, where the function $M^{\uparrow} \geq M$ on $D$ is constructed by averaging of $M$ in rapidly convergent disks as we approach the boundary of the domain $D$ with some possible additive logarithmic component associated with the rate of narrowing of these disks.

Keywords: holomorphic function, sequence of zeros, subharmonic function, Jensen measure, test function, balayage

 Funding Agency Grant Number Russian Science Foundation 18-11-00002 Russian Foundation for Basic Research 18-51-06002

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/faa3597

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UDC: 517.53+517.574+517.987.1
MSC: 30C15, 31A05, 28A25, 30C85
Revised: 12.07.2018
Accepted:04.02.2019

Citation: B. N. Khabibullin, F. B. Khabibullin, “On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems”, Funktsional. Anal. i Prilozhen., 53:2 (2019), 42–58

Citation in format AMSBIB
\Bibitem{KhaKha19} \by B.~N.~Khabibullin, F.~B.~Khabibullin \paper On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems \jour Funktsional. Anal. i Prilozhen. \yr 2019 \vol 53 \issue 2 \pages 42--58 \mathnet{http://mi.mathnet.ru/faa3597} \crossref{https://doi.org/10.4213/faa3597} \elib{http://elibrary.ru/item.asp?id=37298260}