For functions f ∈ L2 (R d, wl (x)dx) with wl 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 Fourier transformsˆftransformsˆ transformsˆf setting by means of kth-order moduli of continuity.