Characterization of bivariate hierarchical quartic box splines on a three-directional grid

• We identify the subspace of “special quartics” generated by quartic box splines.
• We establish the completeness of hierarchical C2C2-smooth quartic box splines.
• We characterize the hierarchical space using “special quartic” polynomials.
• We obtain a basis for the hierarchical spline space over type-I triangulations.

We consider the adaptive refinement of bivariate quartic C2C2-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⊂R2Ω⊂R2 and finite sequence (Gℓ)ℓ=0,…,N(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 C2C2-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.