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

Abstract

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, ∞)×R, which is just the Laguerre hypergroup. Naturally, K n =[0, ∞) n ×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 𝒮ℛ(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.