The Radon transform on ${\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$

D. I. Wallace, Ryuji Yamaguchi
1986 Transactions of the American Mathematical Society  
Let G be SL(2,R). G acts on the upper half-plane U by the Möbius transformation, providing M with the Riemannian metric structure along with the Laplacian, A. We study the integral transform along each geodesic. G acts on P, the space of all geodesies, in a natural way, providing P with its invariant measure and its own Laplacian. (P actually is a coset space of G.) Therefore the above transform can be viewed as a map from a suitable function space on M to a suitable function space on P. We
more » ... e a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms. Introduction. Both the Radon transform and the X-ray transform on a symmetric space arise from the problem of reconstructing a function from its integrals along certain paths. For the Euclidean case, a problem can be stated in one of two ways. Suppose all the integrals of some function / along all straight lines are known. It is then possible to reconstruct / from its line integrals. For R2, the solution to this problem was the inversion of the original Radon-John transform. The ability to invert this particular transform rested in a duality in integral geometry between the points in R2 and the set of lines in R2. Technically, the inversion depends heavily on Fourier analysis on R2. Alternatively, if the space in question is Rn, one might wish to reconstruct / from its integrals over hyperplanes of dimension k. Solutions to both of these problems can be found in Helgason [4, 9] . For the Euclidean case, we shall speak of the Radon transform when we mean an integral over a fc-plane where k < n, and of the X-ray transform when we mean an integral over a straight line. Some references for the Euclidean case include Helgason [5, 9] ,
doi:10.1090/s0002-9947-1986-0849481-7 fatcat:sc3wrxoikvepdctydtjdwhxxwy