A periodicity criterium for quasipolynomials

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 $\beta$.