Smoothness equivalence properties of univariate subdivision schemes and their projection analogues (Q841114)

The author considers the following nonlinear perturbation of a linear subdivision scheme \(S\) that operates on a surface \(M\), embedded in some Euclidean space. First perform linear subdivision in Euclidean space, and then map the obtained data back onto \(M\) via a suitable projection mapping \(P\). Thus, a \(P\)-projection analogue \(T := P \circ S\) of \(S\) is obtained. The author shows that if \(S\) is of \(C^k\) smoothness, then so is \(T\). It is claimed that the results proved here first time confirm a general smoothness equivalence conjecture for noninterpolatory subdivision schemes [\textit{D. L. Donoho}, ``Wavelet type representation of Lie-valued data'', Talk at the IMI ``Approximation and Computation'' meeting May 12--17, 2001, Charleston, South Carolina (2001)]. The method of proximity introduced by Wallner et al. [\textit{J. Wallner}, Constructive Approximation 24, No.~3, 289--318 (2006; Zbl 1106.65017); \textit{J. Wallner} and \textit{N. Dyn}, Comput. Aided Geom. Des. 22, No.~7, 593--622 (2005; Zbl 1083.65023)] is employed for the proof of the author's result.