On Multivariate B-Splines

In this paper multivariate B-splines are constructed in terms of certain multivariate “fundamental solutions” which are analogous to the usual univariate truncated powers. Furthermore, this approach is shown to provide the recurrence relations for the construction of higher order B-splines from lower order ones which were recently found by C. A. Micchelli, A constructive approach to Kergin interpolation in $R^k $ : MultivariateB-splines and Lagrange interpolation, MRC Technical Summary Report 1978. It also allows to relate the smoothness of a B-spline to its knot configuration.