Однородные уравнения типа π-свертки

Abstract

Exponential polynomials satisfying a homogeneous convolution type equation are called its elementary solutions. The traditional solution of a homogeneous equation of the convolution type involves the proof of an approximation theorem that asserts the density of elementary solutions in the set of all solutions of the equation. This article considers homogeneous equations of the π-convolution type in the space of analytic functions on a convex domain, which generalize the well-known homogeneous equations of the convolution type. Equations of a particular form have been considered many times before. Such equations include homogeneous convolution equations, homogeneous q-sided convolution equations, homogeneous q-sided convolution type equations, and homogeneous-convolution equations. In the article the properties of operators of π-convolution type are investigated and an approximation theorem for a homogeneous equation of π-convolution type in an arbitrary convex domain of the complex plane is proved.