Periodic at infinity solutions to differential equations in homogeneous spaces

The article under consideration studies periodic at infinity functions from homogeneous spaces and distributions from harmonic spaces. We give a definition of a homogeneous space of functions defined on the real axis or semi-axis with values in a complex Banach space. For instance, Stepanov spaces, Lebesgue spaces, Wiener amalgams, the space of functions of bounded variation and continuous functions and some of their subspaces belong to this class. On the basis of homogeneous spaces of functions harmonic spaces of distributions are constructed. In the spaces of functions (distributions) under consideration we introduce the concepts of slowly varying and periodic at infinity functions (distributions) and study their properties. For periodic at infinity function (distribution) we also introduce the concepts of canonical and generalized Fourier series, whose coefficients are slowly varying at infinity functions (distributions). We study the properties of Fourier series and derive criteria of periodicity at infinity of a function (distribution). Special attention is paid to obtaining criteria of periodicity at infinity of solutions to differential and difference equations.