The authors investigate interpolation to scattered data by bivariate polynomial natural spline functions on a rectangle. This type of functions is defined as solutions of a special variational problem in Sobolev spaces. As in the general theory of spline interpolation in Hilbert spaces, the existence and uniqueness of the spline interpolant is shown under a certain poisedness assumption.
It is proved that the spline interpolant can be expressed in terms of the bivariate truncated power functions which correspond to the interpolation points. In addition, these interpolants can be computed by solving a system of linear equations which is given explicitly.