Fractional Smoothness in $L^p$ with Dunkl Weight and Its Applications

We define a fractional power of the Dunkl Laplacian, a fractional modulus of smoothness, and a fractional $K$-functional on $L^p$-spaces with Dunkl weight. As an application, we extend our previous results and prove direct and inverse theorems of approximation theory and some inequalities for entire functions of spherical exponential type in the fractional setting.