Wavelet Transform and Radon Transform on the Quaternion Heisenberg Group  

作　　者：Jian Xun HE He Ping LIU 

机构地区：[1]School of Mathematics and Information Sciences,Guangzhou University [2]Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes,Guangzhou University [3]LMAM,School of Mathematical Sciences,Peking University

出　　处：《Acta Mathematica Sinica,English Series》2014年第4期619-636,共18页数学学报（英文版）

基　　金：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);the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.2012000110059)

摘　　要：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 L2（ ）. 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 .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 L2（ ）. 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 .

关 键 词：Quaternion Heisenberg group wavelet transform Radon transform inverse Radon transform 

分 类 号：O152[理学—数学]