Dunkl theory and Jackson inequality in $L_2(\mathbb R^d)$ with power weight

We prove a sharp Jackson inequality in $L_2(\mathbb R^d)$ with the weight $v_k(x)=\prod_{\alpha\in\mathbb R_+}|(\alpha,x)|^{2k(\alpha)}$ defined by the positive subsystem $R_+$ of a finite system of roots $R\subset\mathbb R^d$ and by a function $k(\alpha)\colon R\to\mathbb R_+$ invariant under the reflection group generated by $R$.