Biorthogonal decompositions of the Radon transform by multivariate interpolation

Abstract

The concept of biorthogonal and singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. By application of the theory of multivariate interpolation, e. g. the set of Lagrange polynomials with respect to the space of homogeneous spherical polynomials, we determine new biorthogonal decompositions of the Radon transform. We consider the case of functions with support in the unit ball and the case of functions with support ℝr. In both cases we assume that the functions are square integrable with respect to some weight functions. In the important special case of square integrable functions with respect to the unit ball the structure of the biorthogonal decompositions is easier in comparison with the known singular and biorthogonal decompositions. Especially the calculation of the unknown expansion coefficients can be done by using arbitrary fundamental systems (μ-resolving data set in terms of tomography with a minimum number of nodes) and simplifies essentially. The decompositions are based on a system of zonal (ridge) Gegenbauer (ultraspherical) polynomials which are used in the theory of the Radon transform and in the field of numerical algorithms for the inversion of the transform.