Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension

摘要

For J a planar triangulation, let R-m(r)(J) denote the space of bivariate splines on J such that f is an element of R-m(r)(J) is C-r(tau) smooth across an interior edge tau and, for triangle sigma in J, f vertical bar sigma is a polynomial of total degree at most m(sigma) is an element of Z >= 0. The map m : sigma bar right arrow Z >= 0 is called a non-uniform degree distribution on the triangles in J, and we consider the problem of computing (or estimating) the dimension of R-m(r)(J) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of R-m(r)(J). When all polynomial degrees are sufficiently large, m(sigma) 0, we prove that the number of splines in R-m(r)(J) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013). (C) 2019 Elsevier B.V. All rights reserved.