Smoothness of functions and Fourier coefficients

Abstract

We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the Lp modulus of smoothness, 1 < p < ∞, of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory.