On the Decomposition Problem for Functions of Small Exponential Type

Abstract

The technique of decomposition for functions into the sum or product of two functions is often used to facilitate the study of properties of functions. Some decomposition problems in the weighted Hardy space, Paley-Wiener space, and Bergman space are well known. Usually, in these spaces, functions are represented as the sum of two functions, each of them is "big" only in the first or only in the second quarter. The problem of decomposition of functions has practical applications, particularly in information theory. In these applications, it is often necessary to find those solutions of the decomposition problem whose growth on the negative real semi-axis is "small". In this article we consider the decomposition problem for an entire function of any small exponential type in $\{z:\Re z<0\}$. We obtain conditions for the existence of solutions of the above problem.