On zero sets of weakly localisable pricipal submodules in the Schwartz algebra

We consider the Schwartz algebra $\mathcal P.$ As a linear topological space, it is isomorphic to the space of all distributions compactly supported on the real line. By the Paley — Wiener — Schwartz theorem, the Fourier — Laplace transform establishes the corresponding isomorphism. Submodules of the algebra $\mathcal P$ are defined as closed subspaces which are invariant under the multiplication by the independent variable $z.$ They supply an effective tool to explore the possibility of the spectral synthesis for the differentiation operator in the space $C^{\infty} (\mathbb R).$ In connection with some open questions on the problem of the spectral synthesis in $C^{\infty} (\mathbb R)$, we study principal submodules of the algebra $\mathcal P.$ Earlier, we have obtained the sufficient conditions and the weighted criterion of the weak localisability for principal submodules. These conditions contain some restrictions on the generating function of a submodule. However, one should also consider the following form of the question: knowing the zero set of a principal submodule (or, which is the same, the zero set of its generating function), define whether it is weakly localisable. The complete answer seems to be quite difficult to find. Here, we construct the class of synthesable sequences which are zero sets of weakly localisable principal submodules.