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 inversion of the Radon transform. A singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators AA*, A*A. In this paper we consider a new approach which generalizes this concept. By application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Although 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 which is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The corresponding 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.