Honeycomb and k-fold Hermite subdivision schemes

Along the way, we introduce a general concept called k-fold Hermite subdivision, and analyze its properties with the help of the strong convergence theory of refinement equation. The case of k=2, together with an appropriate symmetry condition, can be used to handle the construction of honeycomb Hermite subdivision schemes. In particular, our framework allows us to construct smoother versions of two interesting honeycomb subdivision schemes in the literature.