Moduli of smoothness and growth properties of Fourier transforms: Two-sided estimates

We prove two-sided inequalities between the integral moduli of smoothness of a function on Rd/TdRd/Td and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann–Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.