Generalized Wavelets And Inversion Of The Radon Transform On The Product Laguerre Hypergroup

Let H1 be the 3-dimensional Heisenberg group. The fundamental manifold ofthe radial function space for H1 can be denoted by [0, +∞)×R, which is just theLaguerre hypergroup(see[25]). Naturally, K~n = [0, +∞)~n×R~n is product Laguerrehypergroup. In this paper, we construct a generalized translation operator on K~n =[0,+∞)~n×R~n, and establish the Plancherel formula on L2(K~n,dμ). Then we discussthe continuous wavelets transform and Radon transform on K~n, and we characterizea subspace SR(K~n) of S (K~n)(Schwartz space), on which Radon transform is abijection. Also, we give another characterized subspace S?,2(K~n) which is equivalentto SR(K~n). Using the inverse wavelet transform, we obtain an inversion formulaof Radon transform on K~n in the weak sense. Analogously, the same case can beextended to the Heisenberg group.