Heisenberg型的群上的Radon变换

The Radon transform on the Heisenberg-type groups

令H={(z,t):z∈Cn,t∈Rm}表示Heisenberg型群,对于(z,t),(z',t')∈H,群乘法法则为(z,t)°(z',t')=(z+z',t+t'+1/2zJz't),其中zJ z't=(z U(1)z't,z U(2)z't,…,z U(m)z't),z't表示z'的转置,U(j)(j=1,2,…,m)是2n×2n反对称实正交矩阵,文章给出了H上的Radon变换,并通过Fourier变换得到了与映射J相关的逆公式.

Let H ={（z,t）：z∈Cn ,t∈Rm}be the Heisenberg-type groups.For （z,t）,（z′,t′）∈H,the group multiplication is given by （z,t）＆#176;（z′,t′）=（z ＋z′,t ＋t′＋12 zJz′t ）,where zJz′t =（zU（1 ）z′t ,zU（2）z′t ,…,zU（m） z′t ）,z′t denotes the transpose of z′,and U（j）（j=1 ,2,…,m）are real skew-symmetric orthogonal matrices.In this paper,we give the definition of the Radon transform on the Heisenberg-type groups,and obtain an inversion formula related with the mapping J by Fourier transform.