Properly distributed subsequence on the line
Aidar Irekovich Abdulnagimov, Александр Сергеевич Кривошеев · Ufimskii Matematicheskii Zhurnal · 2015
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.