Principal submodules in the module of entire functions, which is dual to the Schwartz space, and weak spectral synthesis in the Schwartz space

We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz space $C^{\infty}(a;b)$ are discussed.