Harmonic analysis of periodic at infinity functions in homogeneous spaces

This article is devoted to homogeneous spaces $\mathrm{F}(\mathrm{R},X)$ of functions defined on $\mathrm{R}$ with their values in a complex Banach space X. We introduce a notion of slowly varying at infinity function from $\mathrm{F}(\mathrm{R},X)$. We also consider some criteria for a function to be slowly varying at infinity. Then it is stated that for each slowly varying at infinity function from any homogeneous space (not necessary continuous, for instance, a function from Stepanov space $S^p(R, X)$ or $L^p(R, X)$, ) there exists a uniformly continuous slowly varying at infinity function that differs from the first one by a function decreasing at infinity. In other words, a function from the corresponding subspace $F_0(\mathrm{R}, X)$ . In the second part of the article we introduce a notion of periodic at infinity function from homogeneous space. Our main results are connected with harmonic analysis of periodic at infinity functions from $F(\mathrm{R}, X)$. Periodic at infinity functions appear naturally as bounded solutions of certain classes of differential and difference equations. In this paper we develop basic harmonic analysis for such functions. We introduce the notion of a generalized Fourier series of a periodic at infinity function from homogeneous space. The Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity. Moreover, it is stated that generalized Fourier coefficients of a function that may not be continuous can be chosen continuous. We use methods of the spectral theory of locally compact Abelian group isometric representations (Banach modules over group algebras).