Direct and inverse approximation theorems of functions in the Orlicz type spaces S_M

In the Orlicz type spaces ${\mathcal{S}}_M$, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre $K$-functionals in the spaces ${\mathcal S}_{M}$.