# On Wiener Theorem in Studying Periodic at Infinity Functions with Respect to Subspaces of Vanishing at Infinity Functions

**Струков В. Е., Струкова И. И. On Wiener Theorem in Studying Periodic at Infinity Functions with Respect to Subspaces of Vanishing at Infinity Functions // Taurida Journal of Computer Science Theory and Mathematics, – 2019. – T.18. – №4. – P. 78-91**

https://doi.org/10.37279/1729-3901-2019-18-4-78-91

https://doi.org/10.37279/1729-3901-2019-18-4-78-91

In the article under consideration we study periodic at infinity functions from $C_b(J, X)$, i.e.,

bounded continuous functions defined on the real axis with their values in a complex Banach

space $X$. On the basis of the well-known Wiener theorem we introduce a concept of a set satisfying

Wiener condition. Together with an ordinary subspace $C_0 ⊂ C_b$ 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 gives a function tending to zero at

infinity. Those subspaces we also call vanishing at infinity and denote then as $\mathfrak{C_0}$. So, by choosing

one of the subspaces $\mathfrak{C_0}$ we introduce different types of slowly varying and periodic at infinity

functions (with respect to the chosen subspace).

A function $x \in C_{b,u}$ is called slowly varying at infinity with respect to the subspace $\mathfrak{C_0}$ if

$(S(t)x−x) \in \mathfrak{C_0}$ for all $t \in J$. Respectively, for some $ω > 0$ a function $x \in C_{b,u}$ is called $ω$-periodic

at infinity with respect to the subspace $\mathfrak{C_0}$ if $(S(ω)x − x) ∈ \mathfrak{C_0}$. Nevertheless, these functions

are constructed as extensions of the classes of slowly varying and periodic at infinity functions

respectively, we proved them to be congruent with these classes.

Ordinary periodic at infinity functions appear naturally as bounded solutions of certain

classes of differential and difference equations. So, in our research, we also study the solutions

of differential and difference equations of some kind. It is proved that for those equations, where the right hand side of the equation is a function from any of the subspaces $\mathfrak{C_0}$ of vanishing at

infinity functions, the solutions are periodic at infinity.

The results were received with essential use of isometric representations and Banach modules

theories.

**Keywords**: *Wiener theorem, vanishing at infinity function, slowly varying at infinity function,
periodic at infinity function, Banach space, Banach module, differential equation, difference
equation.*