Pointwise strong (H, Phi) approximation by Fourier series of L^{Psi} integrable functions

We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H; Phi) as an approximation version of the Totik type generalization of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from L^{Psi}. As a measure of such approximation we will use the function constructed by function Psi complementary to Phi on the base of defnition of the L^{Psi} points. Some corollary and remarks will also be given.