On some properties of almost periodic at infinity of functions from homogeneous spaces

In this paper, we consider homogeneous spaces of functions defined on the whole real axis with values in a complex Banach space. A new class of functions almost uniformly periodic at infinity from a homogeneous space is introduced and exmined. Four definitions of such functions are proposed and their equivalence is proved. Fourier series of functions almost periodic at infinity are constructed and their properties are analyzed. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules.