Finitely Generated Submodules in the Module of Entire Functions Determined by Restrictions on the Indicator Function

We study finitely generated submodules in the module $P$ of entire functions bounded by a system of $\rho$-trigonometrically convex weights majorized by a given $\rho$-trigonometrically convex function. Sufficient conditions for the ampleness of a finitely generated submodule in terms of the relative position of the zeros of its generators are obtained. Using these conditions, we prove that each ample submodule in $P$ is generated by two (possibly, coinciding) functions.