On 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 $C_b(\mathbb{J},X),$ i. e. bounded continuous functions defined on an interval $\mathbb{J}=\{\mathbb{R}_{+};\mathbb{R}\}$ with their values in a complex Banach space $X.$ Together with an ordinary subspace $C_0\subset C_b$ of functions vanishing at infinity we define a subspace $(L^1C)_0\subset C_b$ of functions vanishing at infinity upon the average. Then we introduce a range of different subspaces of functions such that $C_0 \subset \mathscr{C}_0 \subset (L^1C)_0$ and call them vanishing at infinity. So, by choosing one of those subspaces $\mathscr{C}_0$ 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 $\mathscr{C}_0$ if $(S(t)x - x)\in \mathscr{C}_0$ for all $t\in\mathbb{J}.$ Respectively, for some $\omega>0$ a function $x\in C_{b,u}$ is called $\omega$-periodic at infinity with respect to the subspace $\mathscr{C}_0$ if $(S(\omega)x - x)\in \mathscr{C}_0.$

Those functions are an extension of the class of periodic at infinity functions, which appear naturally as bounded solutions of certain classes of differential and difference equations. Our main focus is to develop the basic harmonic analysis for periodic at infinity functions (with respect to the chosen subspace $\mathscr{C}_0$) and an analogue of the celebrated Wiener's Lemma that deals with the absolutely convergent Fourier series.

For a periodic at infinity function (with respect to the chosen subspace) we introduce the concepts of canonical and generalized Fourier series with coefficients slowly varying at infinity (not necessarily constant and not necessarily having a limit at infinity) and study their properties. Besides, we prove the summability of the Fourier series by the method of Chesaro.

We also introduce a notion of $\mathscr{C}_0$-invertibility of a continuous function, in terms of which the analogue of Wiener's Lemma is derived: if a $\mathscr{C}_0$-invertible periodic at infinity function with respect to the subspace $\mathscr{C}_0$ has an absolutely convergent Fourier series then each of its $\mathscr{C}_0$-inverse functions also has an absolutely convergent Fourier series.

Moreover, derive a spectral criterium of periodicity at infinity and a criterium of representability of periodic at infinity function with respect to the subspace $\mathscr{C}_0$ as a sum of pure periodic and vanishing at infinity (with respect to $\mathscr{C}_0$) functions.

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