Multivariate divided differences I: basic properties

The notion of the univariate divided differences is generalized to the multivariate case. This generalization is based on a pointwise evaluation of a certain multivariate function. Several properties of the defined multivariate divided difference functional are derived, and a link with the multivariate simplex splines is established. This gives a new generalization of the so-called truncated power function, which is different from the one given in [Multivariate Approximation Theory, Birkhauser Basel, Switzerland, 1979, pp. 69-82] and [SIAM J. Numer. Anal., 17 (1980), pp. 179-191].