proceedings of the american mathematical society Volume 118, Number 2, June 1993 THE RADON TRANSFORM OF BOEHMIANS PIOTR MIKUSINSKI AND AHMED ZAYED (Communicated by Andrew M. Bruckner) Abstract. The Radon transform, which enables one to reconstruct a function of N variables from the knowledge of its integrals over all hyperplanes of di- mension N - 1 , has been extended to Schwartz distributions by several people including Gelfand, Graev, and Vilenkin, who extended it to tempered distribu- tions. In this paper we extend the Radon transform to a space of Boehmians. Boehmians are defined as sequences of convolution quotients and include Schwartz distributions and regular Mikusinski operators. Our extension of the Radon transform includes generalized functions of infinite order with compact support. The technique used in this paper is based on algebraic properties of the Radon transform and its convolution structure rather than on their analytic properties. Our results do not contain nor are contained in those obtained by Gelfand et al. 1. Introduction The Radon transform, which was defined and studied by Radon [11] in 1917, has emerged in the last two decades as one of the most important mathematical tools in various fields of applications in physics, engineering, astronomy, and medicine. The applications of the Radon transform are numerous; however, the most widely known is probably its application in medical imaging, where it plays an essential role in the field of Computerized Axial Tomography or Com- puterized Assisted Tomography, commonly known to the public as CAT scans. The importance of the Radon transform and its inverse lies in the fact that they solve the following reconstruction problem: construct a function f(x, y) given that its integrals over all straight lines are known; or more generally, construct a function fi(xx,... , Xn) given that its integrals over all hyperplanes of dimen- sion N-l are known. To see how this simply stated reconstruction problem plays a central role in a wide range of applications, we refer the reader to [2, Chapter 1] for an excellent exposition. Although in most applications the function / is assumed to have compact support, in theory it is usually assumed to be in the Schwartz space S"(RN) (N > 2) of infinitely differentiable rapidly decreasing functions. The extension Received by the editors October 20, 1991. 1991Mathematics SubjectClassification. Primary 44A12,44A40;Secondary 46F12, 92C55. Key words and phrases. Radon transform, Boehmians, generalized functions. © 1993 American Mathematical Society 0002-9939/93 $1.00 + 1.25 per page 561 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use