In this research, we investigate some approximation theorems for \( L^{2} \) -space on the Laguerre hypergroup which is the fundamental manifold of the radial functions space for the Heisenberg group. An analogue of Bernstein’s theorem is shown. Some inverse theorems of Jackson–Stechkin in terms of best approximations for the moduli of smoothness defined using generalized translation operators on the Laguerre hypergroup are obtained. To prove these theorems, we use functions of bounded spectrum as a tool of approximation.
