On a regularization scheme for linear operators in distribution spaces with an application to the spherical Radon transform

Abstract

This article provides a framework to regularize operator equations of the first kind where the underlying operator is linear and continuous between distribution spaces, the dual spaces of smooth functions. To regularize such a problem, the authors extend Louis method of approximate inverse from Hilbert spaces to distribution spaces. The idea is to approximate the exact solution in the weak topology by a smooth function, where the smooth function is generated by a mollifier. The resulting regularization scheme consists of the evolution of the given data at so called reconstruction kernels which solve the dual operator equation with the mollifier as right-hand side. A nontrivial example of such an operator is given by the spherical Radon transform which maps a function to its mean values over spheres centered on a line or plane. This transform is one of the mathematical models in sonar and radar. After establishing the theory of the approximate inverse for distributions, we apply it to the spherical Radon transform. The article also contains numerical results.