ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

# медленно меняющиеся на бесконечности функции

## The research of some classes of almost periodic at infinity functions

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions.

## Wiener's theorem for periodic at infinity functions

In this article banach algebra of periodic at infinity functions is defined. For this class of functions notions of Fourier series and absolutely convergent Fourier series are introduced. As a result Wiener's theorem analog devoted to absolutely convergent Fourier series for periodic at infinity functions was proved.

## About harmonic analysis of periodic at infinity functions

We consider slowly varying and periodic at infinity multivariable functions in Banach space. We introduce the notion of Fourier series of periodic at infinity function, study the properties of Fourier series and their convergence. Basic results are derived with the use of isometric representations theory.

## Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity

In the paper we introduce and study a new class of almost periodic at infinity functions, which is defined by means of a subspace of  integrally decreasing at infinity functions. It is wider than the class of almost periodic at infinity functions introduced in the papers of A.G.Baskakov (with respect to the subspace of functions vanishing at infinity).