Generalized Lipschitz and Besov spaces in terms of decay of Dunkl transforms in the space \(L^{2}({\mathbb {R}}^{d},w_{l}(x)dx)\)

For functions \(f\in L^{2}\left( {\mathbb {R}}^{d},w_{l}(x)dx \right) \) $f\in L^2\left({\mathbb{R}}^d,,,w_l,(x),d,x\right)$ with \(w_{l}\) $w_l$ is a weight function invariant under the action of an associated Weyl group, we give necessary and sufficient conditions for f to belong to the generalized Lipschitz and Besov spaces in terms of the behaviour of their Dunkl transforms \({\widehat{f}}\) $\widehat{f}$ setting by means of kth-order moduli of continuity.