On fundamental solutions for multivariate singular interpolation

The objective of this paper is to give a constructive approach to the solution of the fundamental functions for cardinal interpolation from a shift-invariant space generated by the (multi-)integer translates of some compactly supported function whose polynomial symbol has a non-empty zero set. This problem was first introduced by Chui, Diamond, and Raphael, where explicit solutions were given for various zero sets. Later, de Boor, Höllig, and Riemenschneider gave an existence proof for zero sets which are more general. In this paper, we give an integral representation of the fundamental solutions that can be made explicit in some cases and we will also give a growth condition of such fundamental solutions. The four-directional box splines will be used as an illustrative example.