Ridge Function Spaces and Their Interpolation Property

We use the concept of "characteristic matrices" to study the problem of interpolating scattered data by ridge functions. We show that the solvability of the interpolation is equivalent to the positive definiteness of the characteristic matrices. On the basis of the theory of characteristic matrices, we give a sufficient condition for the singularity of the interpolation matrices. In a special case, this condition is equivalent to the "closed path" criterion given by Dyn, Light and Cheney (J. Approx. Theory59 (1989), 202-223).