Smoothness Analysis of Subdivision Schemes by Proximity

Linear curve subdivision schemes may be perturbed in various ways, for example, by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes "T" is based on their proximity to the linear schemes "S" which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities which together with C^k smoothness of S imply C^k smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in the case k = 2. The main result of the paper is that convergent geodesic/projection/Lie group analogues of a certain class of factorizable linear schemes have C² limit curves.