Abstract:
Our aim is the approximation of compactly supported probability measures by more elementary measures with “thinner” support. We use a distance between probability measures that is induced by reproducing kernels and their spectral decomposition. For the Euclidean distance kernel on the unit ball, we derive this spectral decomposition from the eigenfunctions and eigenvalues of the polyharmonic operator. The spectral decomposition has effective advantages for numerical minimization. In numerical experiments, we approximate probability measures by measures supported on a finite set of points or supported on closed curves of bounded finite length.