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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 anestimation of the generalized strong mean (H; Phi) as an approximation versionof 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 suchapproximation we will use the function constructed by function Psicomplementary to Phi on the base of defnition of the L^{Psi} points. Somecorollary and remarks will also be given.

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