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). It suffices to turn to the approximation theory for a new class of functions, where the Fourier coefficients are slowly varying at infinity functions with respect to the subspace of functions that decrease integrally at infinity.
