MMCS 2012: Mathematical Methods for Curves and Surfaces pp 383-402 | Cite as
Commutator Estimate for Nonlinear Subdivision
Abstract
Nonlinear multiscale algorithms often involve nonlinear perturbations of linear coarse-to-fine prediction operators S (also called subdivision operators). In order to control these perturbations, estimates of the “commutator” SF − FS of S with a sufficiently smooth map F are needed. Such estimates in terms of bounds on higher-order differences of the underlying mesh sequences have already appeared in the literature, in particular in connection with manifold-valued multiscale schemes. In this paper we give a compact (and in our opinion technically less tedious) proof of commutator estimates in terms of local best approximation by polynomials instead of bounds on differences covering multivariate S with general dilation matrix M. An application to the analysis of normal multiscale algorithms for surface representation is outlined.
Keywords
subdivision operators polynomial reproduction local polynomial best approximation nonlinear multiscale transformsPreview
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