Generalized subdivision and convergence

The subdivision algorithm for box splines over a triangular grid is considered. It is shown that the refined control net, viewed as a piecewise linear approximation converges to the spline surface at the rate of 1/r^2 where r is the degree of refinement. Further generalizations of the subdivision algorithm are given: The corresponding algorithms compute 'control' nets that converge to the surface at the rate of 1/r^3 or 1/r^4 respectively.