  We derive new explicit expressions for the components of Moore-Penrose
inverses of symmetric difference matrices. These generalized inverses are
applied in a new regularization approach for scattered data interpolation based
on partial differential equations. The columns of the Moore-Penrose inverse
then serve as elements of a dictionary that allow a sparse signal
approximation. In order to find a set of suitable data points for signal
representation we apply the orthogonal patching pursuit (OMP) method.
