The algebra of continuous piecewise polynomials

Multivariate piecewise polynomials

This article was supposed to be on `multivariate splines'. An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a `multivariate spline', resulted in the answer that a multivariate spline is a possibly smooth, piecewise polynomial function of several arguments. In particular, the potentially very useful thin-plate spline was thought to belo...

Multivariate Piecewise Polynomials

On the Dimension of Multivariate Piecewise Polynomials

Lower bounds are given on the dimension of piecewise polynomial C 1 and C 2 functions de ned on a tessellation of a polyhedral domain into Tetrahedra. The analysis technique consists of embedding the space of interest into a larger space with a simpler structure, and then making appropriate adjustments. In the bivariate case, this approach reproduces the well-known lower bounds derived by Schum...

Piecewise Polynomials on Polyhedral Complexes

For a d-dimensional polyhedral complex P , the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P, r, k) of degree d. When d = 2 and P is simplicial, in [1] Alfeld and Schumaker give a formula for all three coefficients of f . However, in the polyhedral case, no formula is known. Using loca...

Remarks on Piecewise-linear Algebra

Introduction* A function f: V —>W between real vector spaces is piecewise-linear (PL) if there exists a partition of V into "open polyhedra" X€ (i.e., relative interiors of polyhedra) such that / is affine on each Xt. (As distinct to the case of PL-topology, no continuity is required of /.) Images and preimages under PL-maps give rise to finite unions of open polyhedra, or PL-seίs; conversely P...