Multivariate Hermite–Birkhoff interpolation by a class of cardinal basis functions

A class of cardinal basis functions for Hermite–Birkhoff interpolation to multivariate real functions on scattered data is constructed. The argument is developed first recalling some classical approaches to the multivariate Hermite interpolation problem, and then introducing suitable cardinal basis functions satisfying a vanishing property on the derivatives. A noteworthy special case involving Shepard’s functions is finally discussed, including some numerical examples.