  In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type
inequalities are obtained in terms of best approximations of functions and the
averaged values of their generalized moduli of smoothness. The values of
Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$
are found for classes of periodic functions defined by certain conditions on
the averaged values of the generalized moduli of smoothness.
