Synthesizable sequence and principle submodules in Schwartz module

Abstract

We consider a module of entire functions of exponential type and polynomial growth on the real axis, that is, the Schwarz module with a non-metrizable locally convex topology. In relation with the problem of spectral synthesis for the differentiation operator in the space $C^{\infty} (a;b)$, we study principle submodules in this module. In particular, we find out what functions, apart of products of the polynomials on the generating function, are contained in a principle submodule. The main results of the work is as follows: despite the topology in the Schwarz module is non-metrizable, the principle submodule coincides with a sequential closure of the set of products of its generating function by polynomials. As a corollary of the main result we prove a weight criterion of a weak localizability of the principle submodule. Another corollary concerns a notion of “synthesizable sequence” introduced recently by A. Baranov and Yu. Belov. It follows from a criterion of the synthesizable sequence obtained by these authors that a synthesizable sequence is necessary a zero set of a weakly localizable principle submodule. In the work we give a positive answer to a natural question on the validity of the inverse statement. Namely, we prove that the weak set of a weakly localizable principle submodule is a synthesizable sequence.