О плотности многочленов в специальном пространстве целых функций экспоненциального типа

Abstract

The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.