Harmonic Analysis of Periodic Vectors and Periodic at Infinity Functions

In this paper we study vector-valued slowly varying and periodic at infinity functions of several variables. We introduce the notion of Fourier series and derive an analog of the celebrated Wiener theorem that deals with the absolutely convergent Fourier series. We also derive a criterion of representability of periodic at infinity function as a sum of pure periodic and vanishing at infinity functions and criteria of periodicity at infinity for solutions of difference and differential equations. The main results are derived by means of the spectral theory of isometric group representations.