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 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.

Keywords:

• Heisenberg group
• Laguerre hypergroup
• Radon transform
• Wavelet transform

Acknowledgements

The authors would like to thank the referee for his/her valuable comments on this paper. J. He is supported by the National Natural Science Foundation of China (No. 10371087, 10671041), the Science Research Foundation of Guangzhou Educational Bureau.

Additional information

Notes on contributors

Pei Liu

Email: [email protected]