International Conference on Multivariate Approximation
September 24-27, 2011

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On the dimension of triangular spline spaces

Nelly Villamizar
, Bernard Mourrain
The space of spline functions attached to a subdivided planar domain plays an important role in CAGD and has been considered recently for isogeometric analysis applications. Using homological techniques, we found a lower and an upper bound to the dimension of this spline space, which are more general and give better approximations to the exact value of the dimension than the already existing ones. The formulas allow us to recover the known cases where the lower and upper bounds coincide and also give us some insight on the type of hierarchical subdivision strategy to employ in order to keep this property. The results can be extended to spline spaces on 3-dimensional complexes and applied to any rectilinear subdivision of a polygonal domain, and to mixed splines, which are splines where the order of smoothness may differ on the various edges.