Cardinal interpolation with differences of tempered functions

In this paper, we investigate the existence and uniqueness of cardinal interpolants associated with functions arising from the k^th order iterated discrete Laplacian ^k applied to certain radial basis functions. In particular, we concentrate on determining, for a given radial function Φ, which functions ^kΦ give rise to cardinal interpolation operators which are both bounded and invertible ?₂ (Z³). In addition to solving the cardinal interpolation problem (CIP) associated with such functions ^kΦ, our approach provides a unified framework and simpler proofs for the CIP associated with polyharmonic splines and Hardy multiquadrics.