Commutator Estimate for Nonlinear Subdivision

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.