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In this paper, we give a characterization of Nikol'ski\u{\i}-Besov typeclasses of functions, given by integral representations of moduli ofsmoothness, in terms of series over the moduli of smoothness. Also, necessaryand sufficient conditions in terms of monotone or lacunary Fourier coefficientsfor a function to belong to a such a class are given. In order to prove ourresults, we make use of certain recent reverse Copson- and Leindler-typeinequalities.

On approximations by trigonometric polynomials of classes of functions defined by moduli of smoothness