Smoothness of interpolatory multivariate subdivision in Lie groups

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 Holder exponents.