Harmonic analysis of functions almost periodic at infinity in Banach modules

The article is devoted to homogeneous spaces of functions defined on a locally compact Abelian group and with their values in a complex Banach space. These spaces include a umber of well-known spaces such as the spaces of Lebegue-measurable summable functions, substantially limited functions, bounded continuous functions, continuous vanishing at infinity functions, Stepanov and Holder spaces. It is important that they can be endowed with structure of Banach modules, defined by the convolution of functions. This feature makes it possible to use the concepts and the results of the theories of Banach modules and isometric representations. In the article, we study almost periodic at infinity functions from homogeneous spaces. By using the properties of almost periodic vectors in Banach modules, we study some properties of slowly varying and almost periodic at infinity functions. Two equivalent definitions of an almost periodic at infinity function are introduced, as well as the concept of a Fourier series, which is ambiguous, namely, the Fourier coefficients are defined within the accuracy of a function from the corresponding space vanishing at infinity. We also obtain criteria for a function from a homogeneous space to be slowly varying or almost periodic at infinity. By using the properties of the Beurling spectrum and the concepts of the set of non-almost periodicity of a vector from a Banach module, we obtain a criterion of representability of an almost periodic at infinity function in the form of a sum of vanishing at infinity and the usual almost periodic functions.