Inversion of the Radon transform on the product Laguerre hypergroup by using generalized wavelets

Let H-1 be the three-dimensional Heisenberg group. The fundamental manifold of the radial function space for H-1 can be denoted by [0,infinity) x R, which is just the Laguerre hypergroup. Naturally, K-n = [0,infinity)(n) x R-n is the product Laguerre hypergroup. In this paper, we give the theory of continuous wavelet analysis and the Radon transform on K-n, and devise a subspace S-R(K-n) of S(K-n) (Schwartz space) on which the Radon transform is a bijection. Also, we give two equivalent characterizations on S(K-n) for the Radon transform. By using the inverse wavelet transform we establish an inversion formula of the Radon transform on K-n in the weak sense.