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

In this paper, we give a characterization of Nikol'ski\u{\i}-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1208.612

On Approximations by Trigonometric Polynomials of Classes of Functions Defined by Moduli of Smoothness

In this paper, we give a characterization of Nikol'skiȋ-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to such a class are given. In order to prove our results, we make use of certain recent reverse Copson-type and Leindler-type inequalities