Module and vector space bases for spline spaces (Q808354)

One way of investigating the dimension of and bases for d-variate polynomial spline spaces \(S^ r_ m(\Delta)\) of given smoothness r and polynomial degree m is to view the union \(S^ r(\Delta)=\cup_{m}S^ r_ m(\Delta)\) as a module over the polynomial ring \({\mathbb{R}}[x_ 1,...,x_ d]\). In this paper, it is shown how in certain cases a module basis of \(S^ r(\Delta)\) can be used to find a vector space basis of \(S^ r_ m(\Delta)\) by establishing a relationship between the dimension of \(S^ r_ m(\Delta)\) and the degrees of reduced basis elements of \(S^ r(\Delta)\). Techniques for finding module bases are discussed, especially for cross-cut grids in \({\mathbb{R}}^ 2\).