Multivariate Subdivision Schemes And Divided Differences

A new generalization of divided diierences to the multivariate case is introduced to investigate the regularity of functions produced by subdivision. A general commutation formula is presented for a class of interpolating subdivision schemes. The technique developed is then used to construct subdivision schemes on irregular multilevel meshes. 1. Introduction Subdivision schemes are used to build functions and surfaces. Starting from initial coarse data and following local recursive rules these schemes produce denser and denser sets of points on a curve or surface. Subdivision has been used in geometric modeling, rst as an eecient way for computing splines, and then in a more general context. CC78, DS78, Loo87, Hol, PR98] Functions produced by subdivision also appear as building blocks for wavelets, with corresponding subdivision schemes obtained by repeatedly applying low-pass ltering. Since the functions resulting from subdivision do not generally have any explicit analytic expression , it is important to investigate the convergence of a subdivision process and the regularity of the functions it produces. While Fourier methods and spectral analysis play an important role in the regularity analysis of stationary subdivision CDM91, Dub86, DL92, DLM90, Zor97], subdivision schemes on irregular meshes require a diierent set of tools for their analysis. For instance, the study of univariate irregular subdivision schemes in DGS99] extensively used standard divided diierences, see SB80]. This paper is a natural extension of ideas from DGS99] for the multivariate setting. Whereas there are several generalizations of the deenition of divided diierences in the mul-tidimensional case ((dB95], Kun96], Nea92]), we could not adapt any of them for our speciic purpose. In particular, none of these deenitions could provide us with a local operator deened as

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