On the linear independence of truncated hierarchical generating systems

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of RdRd. Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.