Estimate of K-functionals and modulus of smoothness constructed by generalized spherical mean operator

Abstract

Using a generalized spherical mean operator, we define generalized modu-lus of smoothness in the space \(\mathrm {L}_{k}^{2}(\mathbb {R}^{d})\). Based on the Dunkl operator we define Sobolev-type space 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 for the Dunkl transform on ℝd.