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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 1, Pages 47–52 (Mi ufa131)

A periodicity criterium for quasipolynomials

N. P. Girya, S. Yu. Favorov

V. N. Karazin Kharkiv National University, Kharkiv, Ukraine

Abstract: We consider functions from the $\Delta$ class, which was introduced by M. G. Krein and B. Ja. Levin in 1949. $\Delta$ is a class of almost periodic entire functions of an exponential type with zeros belonging to a horizontal strip of a finite width. In particular, the class contains all finite exponential sums with pure imaginary exponents. Another description of the class $\Delta$ is analytic continuations to the complex plane of almost periodic functions on the real axis with a bounded spectrum such that the infimum and the supremum of the spectrum belong to the spectrum too.
It is proved that any function from the class $\Delta$ with a discrete set of differences of its zeros is a finite product of shifts of the function sin $\sin\omega z$ up to a factor $C\exp\{i\beta z\}$ with real $\beta$.

Keywords: almost periodic function, entire function of an exponential type, zero set, discrete set.

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UDC: 517.518.6

Citation: N. P. Girya, S. Yu. Favorov, “A periodicity criterium for quasipolynomials”, Ufimsk. Mat. Zh., 4:1 (2012), 47–52

Citation in format AMSBIB
\Bibitem{GirFav12} \by N.~P.~Girya, S.~Yu.~Favorov \paper A periodicity criterium for quasipolynomials \jour Ufimsk. Mat. Zh. \yr 2012 \vol 4 \issue 1 \pages 47--52 \mathnet{http://mi.mathnet.ru/ufa131}