Abstract
Let 
be the quaternion Heisenberg group, and let P be the affine automorphism group of 
. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on 
. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on 
. A Semyanistyi-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on 
both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth. In addition, we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on 
.
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The first author is supported by National Natural Science Foundation of China (Grant Nos. 10971039 and 11271091), the second author is supported by National Natural Science Foundation of China (Grant No. 10990012) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 2012000110059)
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He, J.X., Liu, H.P. Wavelet transform and radon transform on the quaternion Heisenberg group. Acta. Math. Sin.-English Ser. 30, 619–636 (2014). https://doi.org/10.1007/s10114-014-2361-y
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DOI: https://doi.org/10.1007/s10114-014-2361-y