Radial wavelet and radon transform on the Heisenberg group

Abstract

Let be the Heisenberg group, and let denote the affine automorphism group of . The theory of continuous wavelet transforms on the Heisenberg group associated with has been studied in the viewpoint of square integral group representations [J.X. He and H.P. Liu, Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane, J. Math. Anal. Appl. 208 (1997), pp. 58–70]. In this paper, we construct a type of radial wavelets on , the Calderón reproducing formula is also valid. In addition, we devise a subspace of Schwartz functions on which the Radon transform is a bijection. Furthermore, we introduce two subspaces of such that the Radon transform and inverse Radon transform hold by using the wavelet transforms. In our new formulae, the inverse Radon transforms are associated with the sub-Laplacian on , and the smoothness on f can be neglected if wavelet functions are differential.

Keywords:

• Radon transform
• wavelet transform
• sub-Laplacian
• inversion formula
• Heisenberg group

AMS Subject Classifications:

• 43A85
• 44A15
• 47G10

Acknowledgements

The authors would like to thank the referee for his or her apt suggestions to this paper. The work for this paper is supported by the National Natural Science Foundation of China (No. 10971039, 11271091).