  We study the local approximation properties in hierarchical spline spaces
through multiscale quasi-interpolation operators. This construction suggests
the analysis of a subspace of the classical hierarchical spline space (Vuong et
al., 2011) 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.
