About four derinitions of almost periodic at infinity functions from homogeneous space

Author(s):  V.E. Strukov, candidate of Sciences, Voronezh State University, Voronezh, Russia, sv.post.of.chaos@gmail.com

I. I. Strukova, candidate of Sciences, Voronezh State University, Voronezh, Russia, irina.k.post@yandex.ru

Issue:  Volume 50, № 3

Rubric:  Mathematics

Annotation:  The article deals with homogeneous spaces of functions defined on the entire real axis (or semi -axis) with their values in a complex Banach space. We introduce and study a new class of almost periodic at infinity functions from homogeneous spaces. Based on the definitions of bounded uniformly continuous periodic at infinity functions we give analogous definitions for functions from homogeneous spaces. The first definition is based on the notion of an -period at infinity, the second one is concerned with the precompactness of the set of shifts, the third one is asymptotic and the fourth is connected to the equiv-alence class from the quotient space being a periodic vector. Then with the help of the properties of periodic vectors we prove the equivalence of all those definitions. The results of the article are obtained with substantial use of isometric representations theory and Banach modules theory

Keywords:  almost periodic at infinity function, homogeneous space, Banach module, almost periodic vector, Beurling spectrum

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