The Generalized Radon Transform In The Dunkl Theory

The Radon transform is one of the important topics in harmonic analysis and integral geometry. It has close connections with the Fourier transform, the Laplacian and the spherical harmonics, so that the methods in harmonic analysis enter into the study for it. It can be applied to solve the the partial differential equations and study geometric objects with some invariance. The microlocal analysis is used to characterize the support theorem of Radon transform and wavelet analysis to the inverse problems.The Dunkl theory is recently developed in last two decades, which contains the framework for special functions with reflection symmetries and generalized Fourier transforms related to root systems, in several variables. The Dunkl theory has close connections with many branch of mathematics, such as root systems, Weyl groups and Hecke algebra and plays an important roles in combinations. The purpose of this thesis is to develop a full treatment on the associated Radon transform,called the Dunkl-Radon transform, in the Dunkl theory. The Dunkl-Radon transform has natural connections with the Dunkl transform, the h-harmonics, and in particular, the rational Dunkl operators and the h-Laplacian.Although unlike other generalizations of the Radon transform on the Euclidean spaces, the Dunkl-Radon transform has no obvious interpretations in geometry, their favorable properties in analysis and potential applications to various problems impel us to study them in a systematic way.·Some tools in the Dunkl theory are studied and extended to the spaces L_K^p(R^d) or L_K^p(S^d-1) in a detailed and clear way. The crux of the matter is that the Lebesgue measurable functions or those in L_K^p(R^d) are not ensured to be measurable in general associated to the representing measures of these operators or operations. This work is important and necessary to the related harmonic analysis as well as the study to the Radon transform. We improves a result of Thangavelu and Xu and proves the boundedness of M_f for all 1≤p<∞.·The Dunkl-Radon transform as well as its dual is introduced. The elementary properties of it are obtained, especially the relations to the Dunkl transform and the Dunkl operators. A support theorem is proved. In particular, the Dunkl-Radon transform and its dual are showed to be symmetry-preserving in the sense of Dunkl theory.·The mixed norm estimate for the Dunkl-Radon transform and the Hardy-Littlewood-Sobolev for the Dunkl-Riesz potential are established, in terms of which the Fuglede formula of Dunkl-Radon transform for functions in L_K^p(R^d) is proved. And then a inversion formula is obtained. The Dunkl-Radon transform is applied to the equation (?)a_jâ–³_h^ju=f and the Cauchy problem of the generalized wave equation associated to the h-Laplacian.·The inversion formula for the basis of the space L_K²(S^d-1×R,(?)_v^-1) is proved, in terms of which the singular value decomposition of R_K is given.·The spherical Dunkl-Radon transform R_K^S, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of h-harmonics.Several inversion formulas of R_K are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.? A generalized Radon transform R_α,β on the plane for functions even in each variable is defined, which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operatorâ–³_α,β and the Jacobi polynomials P_k^β,α(t). The transform R_α,β and its dual R_α,β^* are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R_α,β for functions in L_α,β^p(R₊²) are obtained in terms of the bivariate Hankel-Riesz potential. As a by-product, a new product formula for the Jacobi polynomials is achieved by applying an invariant property of R_α,β.