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.
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Oswald, P., Shingel, T. (2014). Commutator Estimate for Nonlinear Subdivision. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_22
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DOI: https://doi.org/10.1007/978-3-642-54382-1_22
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