Principal Submodules in the Schwartz Module

Abstract

In this paper we consider the Schwartz module of entire functions of exponential type having polynomial growth along the real axis. This module is equipped with the non-metrizable locally convex topology. We establish that any principal submodule is the set of limits of converging sequences which members are polynomials multiplied by the generator of the submodule. We also obtain a weight weak localizability criterion for principal submodules and some results concerning the notion «synthesizable sequence» recently introduced by A. Baranov and Yu. Belov.