  In weighted Orlicz type spaces ${\mathcal S}_{_{\scriptstyle \mathbf
p,\,\mu}}$ with a variable summation exponent, the direct and inverse
approximation theorems are proved in terms of best approximations of functions
and moduli of smoothness of fractional order. It is shown that the constant
obtained in the inverse approximation theorem is in a certain sense the best.
Some applications of the results are also proposed. In particular, the
constructive characteristics of functional classes defined by such moduli of
smoothness are given. Equivalence between moduli of smoothness and certain
Peetre $K$-functionals is shown in the spaces ${\mathcal S}_{_{\scriptstyle
\mathbf p,\,\mu}}$.
