Article
Ufa Mathematical Journal
Volume 4, Number 1, pp. 43-47
Criterium of periodicity for quasipolynomials.
Favorov S.Yu., Girya N.P.
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We consider functions from the class $\Delta$, which introduced by M.G. Krein and B.Ja. Levin in 1949. $\Delta$ is the class of almost periodic entire functions of exponential type with zeros into 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 bounded spectrum such that infimum and supremum of the spectrum belong to the spectrum too. It is proved in the article 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 $\sin \omega z$, up to a factor $C\exp\{i\beta z\}$ with real $\beta$.