Subdivision surfaces as an extension of splines have become a promising technique for addressing PDEs on models with complex topologies in isogeometric analysis. This has sparked interest in exploring the approximation by subdivision function spaces. Quasi-interpolation serves as a significant tool in the field of approximation, offering benefits such as low computational expense and strong numerical stability . In this paper, we propose a straightforward approach for constructing the quasi-interpolation projectors of subdivision function spaces that features explicit formulations and achieves a highly desirable approximation order. The local interpolation problem is constructed based on the subdivision mask and the limit position mask, overcoming the cumbersome evaluation of the subdivision basis functions and the difficulty associated with deriving explicit solutions to the problem. Explicit quasi-interpolation formulas for the loop, modified loop, and Catmull–Clark subdivisions are provided. Numerical experiments demonstrate that these quasi-interpolation projectors achieve an expected approximate order and present promising prospects in isogeometric collocation.

Box splines are considered as a natural generalization of univariate uniform B-splines. Box splines have local support and are positive in the interior of its support. The translates of box splines form a partition of unity. In this paper we extend the hierarchical paradigm of tensor product B-splines to box splines, which are called hierarchical box splines. We take the application of quartic smooth hierarchical box splines in surface fitting as an example to demonstrate the adaptivity and flexibility of hierarchical box splines.