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Abstract

We consider the adaptive refinement of bivariate quartic C2-smooth box spline spaces on the three-directional (type-I) grid G . The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a bounded domain Ω⊂R2 and finite sequence (Gℓ)ℓ=0,…,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis.

6. Conclusion

We extended the discussion of the completeness of hierarchical spline spaces from Mokriš et al. (2014) to the case of hierarchies of bivariate quartic C2-smooth box splines on type-I triangulations. There are two main differences to the original approach, which was formulated for tensor-product splines. First, since box splines do not span the whole space of quartic polynomials, a special polynomial subspace – the special quartics – had to be introduced. In some sense this situation generalizes the tensor-product case, where the B-splines span a polynomial space of a given (coordinate-wise) bi-degree, instead of the space of bivariate polynomials of a given total degree.