Refinable spaces and local approximation estimates for hierarchical splines

We study the local approximation properties in hierarchical spline spaces through multiscale quasiinterpolation operators. This construction suggests the analysis of a subspace of the classical hierarchical spline space (Vuong et al. (2011) A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng., 200, 3554-3567) which still satisfies the essential properties of the full space. The B-spline basis of such a subspace can be constructed using parent-children relations only, making it well adapted to local refinement algorithms.