Convolution Equations in spaces of Entire Functions of Exponential Type

Abstract

In this paper, we present certain results on convolution equations and shift-invariant subspaces in special spaces of entire functions of exponential type. Problems related to a convolution equation and invariant subspaces were considered in various spaces of holomorphic functions (see, e.g., the surveys [1, 3] and the books [4, 5]), and the choice of the given spaces of holomorphic functions is related to the fact that in many other spaces of holomorphic functions, there arise difficulties connected with the harmonic analysis and synthesis, i.e., with the existence of invariant subspaces in which, as was proved by D. I. Gurevich [10], either there are no polynomial–exponential element, or they are not dense in a subspace. In the chosen class, there are none of these initial difficulties. 1. Space of Entire Functions PG Let G be a Runge domain in the space C n , O(G) be the space of functions holomorphic in G equipped with the natural topology, i.e., the topology of the uniform convergence on all compact sets in G. Denote by O � (G) the dual space to O(G), i.e., the space of all linear continuous functionals on O(G )e quipped with the strong topology. Sometimes, elements of T ∈O � (G) are called analytic functionals on the domain G. The space O(G) is a closed subspace of the space C 0 (G) of all functions continuous in G, and, therefore, for every analytic functional T ∈O � (G), there exists a measure µ, supp µ G, such that