Effortless quasi-interpolation in hierarchical spaces

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces, which are composed of a hierarchy of nested spaces. The hierarchical quasi-interpolants are described in terms of the truncated hierarchical basis. Once for each level in the hierarchy a quasi-interpolant is selected in the corresponding space, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties of the quasi-interpolants selected at each level are preserved in the hierarchical construction. In particular, hierarchical local projectors are constructed, and the local approximation order of the underling hierarchical space is investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.