Canonical representation of piecewise-polynomial functions with nondegenerate linear-domain partitions

Piecewise-linear (PWL) functions are a widely used class of nonlinear approximate functions with applications in both mathematics and engineering. As an extension of this function class, piecewise-polynomial (PWP) functions with a linear-domain partition represents a more general function class than PWL functions, in that all function pieces in a partitioned domain are (instead of hyperplanes) hypersurfaces described by polynomials. Just like PWL functions, the global expression of PWP functions requires a so-called canonical representation, which is meaningful for practical applications. However, such a canonical representation is still unknown. Our study showed that it is not a straightforward extension of the canonical representation of PWL functions; instead, it has a more general form than the latter. In this paper, we discuss the canonical representation of PWP functions with nondegenerate linear-domain partitions. A canonical representation formula is derived and a sufficient condition for its existence is given. We show that under some degree constraints, the derived canonical formula reduces to the canonical formula of PWL functions. The consistent variation property, which is a sufficient and necessary condition for the canonical representation of PWL functions, is found to be less important for PWP functions