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

We consider the adaptive refinement of bivariate quartic C-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 finite sequence (G)l=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting cells from each level such that their closure covers the entire domain Ω, which is a bounded subset of R. A suitable selection procedure allows to define a basis spanning a hierarchical box spline space. As our main result, we derive a characterization of this space. More precisely, under certain mild assumptions on hierarchical grid, the hierarchical spline space is shown to contain all C-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials.