On Distributions That Are Almost Periodic at Infinity

Abstract

This paper is devoted to the study of slowly varying and almost periodic at infinity distributions from harmonic spaces; a number of spaces of homogeneous functions are considered. The notion of a harmonic space of distributions is introduced; this space is constructed by a homogeneous functional spaces. Properties of harmonic spaces of distributions endowed with the structure of Banach modules are studied. We prove that such spaces are isometrically isomorphic to the corresponding homogeneous functional spaces. Based on the definitions of slowly varying and almost periodic at infinity functions from a homogeneous space, we introduce the notions of slowly varying and almost periodic at infinity distributions from a harmonic space. Using methods of abstract harmonic analysis, we construct Fourier series of almost periodic distributions at infinity and examine their properties. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules.