THE RADON TRANSFORM ON Z

where S + x denotes the setf-s + x i s e S ) . Thus, the Radon transform can be thought of as a way of replacing/by a "smeared out" version of /. This form of the transform represents a simplified model of the kind of averaging which occurs in certain applied settings, such as various types of tomography and recent statistical averaging techniques. A fundamental question which arises in connection with the Radon transform is whether or not it is possible to invert it, i.e., whether one can recover (in principle) the function/from knowledge of Fs. In this paper we investigate this problem in detail for several special classes of groups, including the group of binary ^-tuples under modulo 2 addition.