  In this paper we study direct and inverse approximation inequalities in
$L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these
estimates in their sharp form substantially improving previous results. We also
establish new estimates of the modulus of smoothness of a function $f$ via the
fractional powers of the Dunkl Laplacian of approximants of $f$. Moreover, we
obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl
transforms. Needed Pitt-type and Kellogg-type Fourier--Dunkl inequalities are
derived.
