Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

Author： Grohs Philipp  

Publisher： Springer Publishing Company

ISSN： 0029-599X

Source： Numerische Mathematik, Vol.113, Iss.2, 2009-08, pp. : 163-180

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Abstract

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. (Constr Approx 24(3):289-318, 2006; Comput Aided Geom Design 22(7):593-622, 2005). 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.

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