On $2$-generateness of weakly localizable submodules in the module of entire functions of exponential type and polynomial growth on the real axis

Abstract

In the work we consider a topological module P(a; b) of entire functions, which is the isomorphic image under the Fourier-Laplace transform of the Schwarz space of distributions with compact supports in a finite or infinite interval (a; b) ⊂ R. We prove that each weakly localizable module in P(a; b) is either generated by its two elements or is equal to the closure of two submodules of special form. We also provide dual results on subspaces in C∞(a; b) invariant w.r.t. the differentiation operator.