Nonlinearly Weighted L1-Solvability of the First Kind Fredholm Integral Equation

The Fredholm integral equation of the first kind is a well known example of linear ill-posed problems and of functional equations that do not admit of a direct analytical solution in terms of their data. We show that a variational perturbation of the kernel, K ¨, x., ¨, x.gV, in an inverse-problem space, dual to the solution space, can yield for the integral equation a maximum a posteriori direct primal solution c x., xgI. This dual space L1C V; g K..happens to be a certain restriction of a nonlinearly covariantly.weighted L1-space. It is proved that the emerging analytical primal solution must satisfy an equivalent regularizing Urysohn nonlinear integral equation, with a movable bifurcation point, and that it belongs also in a certain nonlinearly weighted L1 I;r c..space invoked by L1C V; g K... An analytical explanation is advanced for a possible autotransformability between the first-kind and second-kind Fredholm integral equations as a result of perturbations of K only in the dual space.