The approximation of functions from L log L ( log log L ) ( S N ) by Fourier-Laplace series

In this paper, a study of the approximation of functions from L log L ( log log L ) ( S N ) by Fourier-Laplace series is performed. It is proved that the maximal operator of the Riesz means of the Fourier-Laplace series is bounded, from L 1 to L log L ( log log L ) ( S N ) . The result provides a natural and intrinsic characterization of the approximation of the functions by Fourier-Laplace series.