Wiener theorem for studying almost periodic at infinity functions

The article under consideration is devoted to some problems of harmonic analysis of continuous slowly varying and almost periodic at infinity functions. We consider a number of various subspaces of continuous functions disappearing at infinity. On the basis of the well-known Wiener theorem we introduce a concept of a set satisfying Wiener condition. We consider various subspaces of continuous functions vanishing at infinity in different senses, not necessarily tending to zero at infinity. For example, integrally vanishing at infinity functions and functions whose convolution with any function from the set satisfying Wiener condition give a function tending to zero at infinity. Then we construct the spaces of slowly varying and periodic at infinity functions with respect to any of those subspaces. The constructed spaces are proved to coincide with the ordinary spaces of slowly varying and periodic at infinity functions respectively (regardless of the choice of a subspace). For functions, almost periodic at infinity ( with respect to a subspace) four various definitions are given and their equivalence is proved. The results are applied to the research of the properties of differential equations solutions. The results of the article are received with essential use of methods of isometric representations and Banach modules theories.