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

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.