In this work, we introduce the β -semigroup for β>0, which unifies and extends the classical Poisson (for β=1) and heat (for β=2) semigroups within the Dunkl analysis framework. Leveraging this semigroup, we derive an explicit representation for the inverse of the Dunkl–Riesz potential and characterize the image of the function space Lkp(Rn) for 1≤p0. To analyze the convergence of these approximations, we introduce the concept of η -smoothness at a point x0 in the Dunkl setting. We show that if a function f∈Lkp(Rn)∩Lk2(Rn), for 1≤p≤∞, possesses η -smoothness at x0, then the truncated hypersingular approximations converge to f(x0) as ϵ→0+.