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IMA Journal of Numerical Analysis Advance Access published online on March 25, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn065
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Gang Xie{dagger}

Department of Mathematics, East China University of Science and Technology, Shanghai, China 200237

Thomas P. -Y. Yu{ddagger}

Department of Mathematics, Drexel University, 3141 Chestnut Street, 206 Korman Center, Philadelphia, PA 19104, USA

{dagger} Email: gangxie2006{at}gmail.com

{ddagger} Corresponding author. Email: yut{at}drexel.edu

Received on 13 August 2007. Revised on 3 September 2008.


  Abstract

We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme Formula with the log–exp adaption to Lie-group-valued data in Ur Rahman et al. (2005, Multiscale Model. Simul., 4, 1201–1232) produces parameterized curves on the Lie group that are as smooth as the smoothness of Formula —no matter how smooth Formula is. We present both an extrinsic proof and an intrinsic proof. We discuss two variations of our main result. (i) We illustrate how smoothness equivalence can break down in a variant of the original log– exp scheme. (ii) We show that the main result of this paper can be easily extended to a multivariate setting.

Key Words: subdivision schemes; Lie groups; matrix Lie groups; interpolation; smoothness equivalence; breakdown of smoothness equivalence


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