%0 Journal Article
%T The Radon Transforms on the Generalized Heisenberg Group
%A Tianwu Liu
%A Jianxun He
%J ISRN Mathematical Analysis
%D 2014
%R 10.1155/2014/490601
%X Let be the generalized Heisenberg group. In this paper, we study the inversion of the Radon transforms on . Several kinds of inversion Radon transform formulas are established. One is obtained from the Euclidean Fourier transform; the other is derived from the differential operator with respect to the center variable . Also by using sub-Laplacian and generalized sub-Laplacian we deduce an inversion formula of the Radon transform on . 1. Introduction In the past decade the research of Radon transform on the Euclidean space has made considerable progress due to its wide applications to partial differential equations, X-ray technology, radio astronomy, and so on. The basic theory and some new developments can be found in [1] by Helgason and the references therein. The combination of Radon transform and wavelet transform has proved to be very useful both on pure mathematics and its applications. Therefore, it is very meaningful to give the inversion formula of the Radon transforms by using various ways. The first result in the area is due to Holschneider who considered the classical Radon transform on the two-dimensional plane (see [2]). Rubin in [3, 4] extended the results in [2] to the -dimensional Radon transform on and totally geodesic Radon transforms on the sphere and hyperbolic space. Heisenberg group is a vital Lie group with the underlying . Strichartz [5] discussed the Radon transform on the Heisenberg group. Nessibi and Trim¨¨che [6] obtained an inversion formula of the Radon transform on the Laguerre hypergroup by using the generalized wavelet transform. Afterwards, He and Liu studied the analogous problems on the Heisenberg group and Siegel type Lie group (see [7, 8]), and Rubin [9] achieved some new progress of the Radon transform on . In [10] the authors gave the definition of generalized Heisenberg group denoted by and dealt with some problems related to geometric analysis. In this paper, we investigate the inversion formulas of the Radon transform on the generalized Heisenberg group. From the Euclidean Fourier transform and group Fourier transform, we deduce inversion formulas of the Radon transform on associated with differential operators and generalized sub-Laplacian. Let be an -dimensional vector, where are positive real constants for . We can turn into a non-Abelian group by defining the group operation as This group is called the generalized Heisenberg group and is denoted by . It is obvious that the generalized Heisenberg group becomes ordinary Heisenberg group if all for . For any -dimensional vectors , , we define For , , the
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2014/490601/