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We define fractional power of the Dunkl Laplacian, fractional modulus ofsmoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight.As application, we prove direct and inverse theorems of approximation theory,and some inequalities for entire functions of spherical exponential type infractional settings.

Fractional smoothness in $L^p$ with Dunkl weight and its applications