On the dimension of spline spaces on planar T-subdivisions

Details On the dimension of spline spaces on planar T-subdivisions On the dimension of spline spaces on planar T-subdivisions

2010-11-08

arxiv.org/abs/1011.1752v1

Mathematics

Algebraic Geometry

Scientific paper

We analyze the space S_{m, m'}^{r, r'} (T) of bivariate functions that are piecewise polynomial of bidegree (m, m') and class C^{r, r'} over a planar T-subdivision. We give a new formula for the dimension of this space by exploiting homological techniques. We relate this dimension to the number of nodes on the maximal interior segments of the subdivision, give combinatorial lower and upper bounds on the dimension of these spline spaces for general hierarchical T-subdivisions. We show that these bounds are exact, for high enough degrees or if the subdivision is enough regular. Finally, we analyse cases of small degrees and regularities.

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