  Exact Jackson-type inequalities are obtained in terms of best approximations
and averaged values of generalized moduli of smoothness in the spaces
${\mathcal S}^p$. The values of Kolmogorov, Bernstein, linear, and projective
widths in the spaces ${\mathcal S}^p$ are found for classes of periodic
functions defined by certain conditions on the averaged values of the
generalized moduli of smoothness.
