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

In a Hilbert space L2,α:= L2(ℝ, |x|2α+1dx), α > − 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 L2,α. Based on 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.