Equivalence of $K$-functionals and moduli of smoothness constructed by generalized Dunkl translations

In a Hilbert space $L_{2,\alpha}:=L_2(\mathbb{R},|x|^{2\alpha+1}dx)$, $\alpha>-1/2$, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space $L_{2,\alpha}$. On the base of the Dunkl operator we define Sobolev-type spaces and $K$-functionals. The main result of the paper is the proof of the equivalence theorem for a $K$-functional and a modulus of smoothness.