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IMA Journal of Numerical Analysis Advance Access originally published online on July 31, 2008
IMA Journal of Numerical Analysis 2009 29(3):760-772; doi:10.1093/imanum/drn040
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Smoothness of interpolatory multivariate subdivision in Lie groups

Philipp Grohs{dagger}

Institute of Geometry, Graz University of Technology, Kopernikusgasse 24/IV, A 8010 Graz, Austria

{dagger} Email: pgrohs{at}tugraz.at, grohs{at}geometrie.tuwien.ac.at

Received on 23 April 2007. Revised on 14 January 2008.


  Abstract

Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold-valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log–exp analogy. The main result of the paper is that a multivariate interpolatory Lie-group-valued subdivision scheme derived from a linear scheme is at least as smooth as the linear scheme, where smoothness is understood in terms of Hölder exponents.

Key Words: subdivision; nonlinear subdivision; multivariate subdivision; Lie group; Lie group subdivision; Hölder exponents; smoothness equivalence


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