Multivariate splines and polytopes

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-Béz...

Graphs, Syzygies, and Multivariate Splines

The module of splines on a polyhedral complex can be viewed as the syzygy module of its dual graph with edges weighted by powers of linear forms. When the assignment of linear forms to edges meets certain conditions, we can decompose the graph into disjoint cycles without changing the isomorphism class of the syzygy module. Thus we can use this decomposition to compute the homological dimension...

Multivariate Adaptive Regression Splines

Multivariate Differences, Polynomials, and Splines

A constructive approach to Kergin interpolation in IR k : multivariate B-splines and La-19 Corollary 4.19. Let S be a nite set in IR d and let H be the set of distinct elements from the directional matrix N. Then N (S) = X H () (S): That is, if f is a function deened on S, then there exists a polynomial in N agreeing with f on S if and only if there exist polynomials p in () such that P p agree...

Multivariate Differences, Polynomials, and Splines

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The relationship between such a linear functional and polynomial interpolation resembles that between the divided difference and Lagrange interpolation. Applying the functional to the shifted multivari...