Multivariate Splines with Nondegenerate Partitions

Any set of hyperplanes partitions Eⁿ into a set of polyhedra. A multivariate spline of degree n is a polynomial of total degree n on each polyhedron with all partial derivatives of order $n - 1$ being continuous everywhere. An especially simple canonical form is presented for splines with respect to nondegenerate (if a set of hyperplanes has nonempty intersection then the corresponding set of normal vectors is linearly independent) partitions. Use of the canonical form, for fitting data, involves linear regression for fixed partitions and nonlinear regression for varying partitions. The canonical form gives rise to an ill-conditioned linear regression problem. However, some preliminary numerical experience in low dimensions indicates that the ill-conditioning is overcome with the use of singular value decomposition.