Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis

In the work we consider a topological module of entire functions $\mathcal P(a;b)$, which is the isomorphic image of Fourier–Laplace transform of Schwarz space formed by distributions with compact supports in a finite or infinite segment $(a;b)\subset\mathbb{R}$. We study the conditions ensuring that the principal submodule of module $\mathcal P(a;b)$ can be uniquely recovered by the zeroes of a generating function.