On the linear independence of (truncated) hierarchical subdivision splines

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, (truncated) hierarchical generating systems for nested spaces of spline
functions defined on domains in R
d have been recently introduced. Their 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 to spline functions that are defined
on domain manifolds, in particular focusing on the case of subdivision splines generated
by the Catmull-Clark, Loop, and modified Butterfly subdivision schemes. Since the property of local linear independence is no longer available, we introduce the concept of safe
subdomains, which allows us to guarantee linear independence. We provide a catalog of
safe subdomains that facilitates the design of domain hierarchies with linearly independent
(truncated) hierarchical generating systems.