Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Sherstyukov V. B. The problem of Leont'ev on entire functions of completely regular growth. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 2, pp. 30-35. DOI: 10.18500/1816-9791-2013-13-2-1-30-35, EDN: SJJAWV

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
27.02.2013
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Russian
Heading: 
Mathematics
UDC: 
517.547.2
DOI: 
10.18500/1816-9791-2013-13-2-1-30-35
EDN: 
SJJAWV

The problem of Leont'ev on entire functions of completely regular growth

Autors: 
Sherstyukov Vladimir Borisovich, National Research Nuclear University MEPhI
Abstract: 

We consider an entire function of exponential type with all its zeros are simple and form a sequence with the index condensation zero. On the set of zeros a function of its derivative is growing quickly. Required to determine whether original function have complete regularity of growth. This problem, which arose in the theory of representation of analytic functions by exponential series was posed by A. F. Leontiev more than forty years ago and has not yet been solved. In this paper we show that the aforesaid problem a positive solution if the function is “not too small” on a straight line. 

Key words: 
Banach space
integral operator
Bochner–Phillips theorem
Fourier series of an operator
inverse-closedness of a subalgebra
Wiener's pair of algebras
References: 
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Received: 
13.08.2012
Accepted: 
11.01.2013
Published: 
27.02.2013
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