The Radon Transform on SL(2,R)/SO(2,R)

Abstract

Let G be SL(2,R). G acts on the upper half-plane U by the Mobius 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 prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms. 1. Introduction. Both the Radon transform and the X-ray transform on a symmetric space arise from the problem of reconstructing a function from its inte- grals 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 geom- etry 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 dimen- sion 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), Strichartz (17), Radon (13), Peters (12), Solmon (15, 16), and Smith, Solmon and Wagner (14). When one wishes to define the analogous transform on a general symmetric space, several options become available. The source of these options lies in the generalization of the notion of hyperplane. In R2, for example, there are two ways of thinking of an (n — l)-dimensional hyperplane. One is as a n — 1 submanifold. The term totally geodesic means that a straight line tangent to a particular plane is actually contained in it. Thus, one generalization of the Radon transform which has been exploited is the non-Euclidean case of integrals over submanifolds of dimension k. A theory, including inversion formula, has been worked out for the case where k is even, in Helgason (9). More can be found in Gel'fand et al. (2), and Lax and Phillips (11).