Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

We study the following modification of a linear subdivision scheme S: Let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M . Then the P -projection analogue of S is defined as T := P ◦ S. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et. al. [10, 9]. While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.