On the null space of a class of Fredholm integral equations of the first kind

We investigate the null space of Fredholm integral operators of the first kind with TD := integral B D(x)k(x, center dot) dx, where B is a ball, the integral kernel satisfies k(x, y) = Sigma(infinity)(n=0) c(n) vertical bar x vertical bar(ln)/vertical bar y vertical bar(n-2+q) P-n((q)) (x/vertical bar x vertical bar center dot y/vertical bar y vertical bar), x is an element of B, y is an element of R-q \ B, where (c(n)) and (l(n)) are sequences with particular constraints, and the P-n((q)) are Gegenbauer polynomials. We first discuss the case of a 3-dimensional ball in detail, where the P-n((3)) = P-n are Legendre polynomials, and then derive generalizations for the q-dimensional ball. The discussed class includes some important tomographic inverse problems in the geosciences and in medical imaging. Amongst others, uniqueness constraints are proposed and compared. One result is that information on the radial dependence of D is lost in TD. We are also able to generalize a famous result on the null space of Newton's gravitational potential operator to the R-q. Moreover, we characterize the orthonormal basis of the derived singular value decomposition of T as eigenfunctions of a differential operator and as basis functions of a particular Sobolev space. Our results give further insight to the interconnections of magnetic field inversion on the one side and gravitational/electric field inversion on the other side.