This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in Toshniwal et al. (2017). The approach in Toshniwal et al. (2017) breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. (2017) yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.

For T a planar triangulation, let R_m ^r(T) denote the space of bivariate splines on T such that f∈R_m ^r(T) is C^r(τ) smooth across an interior edge τ and, for triangle σ in T, f|_σ is a polynomial of total degree at most m(σ)∈Z_≥0. The map m:σ↦Z_≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of R_m ^r(T) 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(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in R_m ^r(T) 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).