BIORTHOGONAL DECOMPOSITIONS OF THE RADON-TRANSFORM

The concept of singular value decompositions is a valuable tool in the examination of ill-posed inverse problems Af = g such as the inversio n of the Radon transform. A singular value decomposition depends on th e determination of suitable orthogonal systems of eigenfunctions of th e operators AA, A*A. In this paper we consider a new approach which g eneralizes this concept. By application of biorthogonal instead of ort hogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Al though it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator. With respect to the Radon transform for functions with support in the unit ball we apply the system of Appell polynomials whi ch is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The correspondi ng biorthogonal decompositions show some advantages in comparison with the known singular value decompositions. Vice versa by application of our decompositions we are able to prove new properties of the Appell polynomials.