This article is cited in 5 papers

Properly distributed subsequence on the line

A. I. Abdulnagimov^a, A. S. Krivoshyev<sup>b

a</sup> Ufa State Aviation Technical University, Karl Marx str., 12, bld. 1, 450000, Ufa, Russia
^b Institute of Mathematics CC USC RAS, Chernyshevsky str., 112, 450008, Ufa, Russia

Abstract: In the article we consider first order sequences of complex numbers. We prove that a sequence of nonzero minimal density contains a subsequence of the same density. We also prove that a real sequence of nonzero minimal density contains a properly distributed subsequence. Basing on this fact, we prove a result on representation of an entire function of exponential type with real zeros as a product of two entire functions with the same properties. Moreover, one of these functions has a regular growth. As a corollary, we obtain a result on completeness of exponential systems with real exponents in the space of analytic functions in a bounded convex domain of the complex plane.

Keywords: entire function, regular growth, zero set.

UDC: 517.5

MSC: 30D10

Received: 09.07.2014

 English version: Ufa Mathematical Journal, 2015, 7:1, 3–12

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