  We prove two-sided inequalities between the integral moduli of smoothness of
a function on $\mathbb{R}^d/\mathbb{T}^d$ 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.
