Module and vector space bases for spline spaces

Abstract

For Δ, a triangulated d-dimensional region in R d , let S m r ( Δ) be the vector space of all C r functions F on Δ such that for any simplex σ ϵ Δ, F¦ σ is a polynomial of degree at most m. S m r ( Δ) is the often studied vector space of splines on Δ of degree m and smoothness r. We define S r ( Δ) = ∨ m S m r ( Δ). S r ( Δ) is a module over the polynomial ring R [ x 1,…, x d ]. In certain cases a module basis for S r ( Δ) provides vector space bases for the corresponding S m r ( Δ) via simple linear algebra. In this work we examine that relationship and consider techniques for finding module bases of spaces S r ( Δ). A basis for S r ( Δ) is reduced if every element F in S r ( Δ) can be represented using only basis elements of degree less than the degree of F. We show the relationship between the dimension of the spaces S m r ( Δ) and the degrees of the reduced basis elements of S r ( Δ). Ths result leads to techniques for finding module bases. These techniques are used to find module bases for spline spaces on cross-cut grids.