On the almost periodic at infinity functions from homogeneous spaces

We consider homogeneous spaces of functions defined on the real axis (or semi-axis) with values in a complex Banach space. We study the new class of almost periodic at infinity functions from homogeneous spaces. The main results of the article are connected to harmonic analysis of those functions. We give four definitions of an almost periodic at infinity function from a homogeneous space and prove them to be equivalent. We also introduce the concept of a Fourier series with slowly varying at infinity coefficients (neither necessarily constant nor necessarily having a limit at infinity). It is proved that the Fourier coefficients of almost periodic at infinity function from a homogeneous space (not necessarily continuous) can be chosen continuous. Moreover, they can be extended on $\mathbb{C}$ to bounded entire functions of exponential type. Besides, we prove the summability of Fourier series by the method of Bochner–Fejer. The results were received with essential use of isometric representations and Banach modules theory.