Abstract
We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme S 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 S—no matter how smooth S 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.
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