Systems of convolution equations in complex domains

Abstract

In this paper we study the systems of convolution equations in spaces of vector-valued functions of one variable. We define an analogue of the Leontiev interpolating function for such systems, and we provide a series of the properties of this function. In order to study these systems, we introduce a geometric difference of sets and provide its properties. We prove a theorem on the representation of arbitrary vector-valued functions as a series over elementary solutions to the homogeneous system of convolution equations. These results generalize some well-known results by A.F. Leontiev on methods of summing a series of elementary solutions to an arbitrary solution and strengthen the results by I.F. Krasichkov-Ternovskii on summability of a square system of convolution equations. We describe explicitly domains in which a series of elementary solutions converges for arbitrary vector-valued functions. These domains depend on the domains of the vector-valued functions, on the growth of the Laplace transform of the elements in this system, and on the lower bound of its determinant. We adduce examples showing the sharpness of this result. Similar results are obtained for solutions to a homogeneous system of convolution equations, and examples are given in which the series converges in the entire domain of a vector-valued function.