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Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

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In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function  which has sufficient regularity and which is fundamental for interpolation [that means, (0)=1 and (α)=0 for all α∈Zs∖{0}].

Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function . Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.

A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.
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Keywords: Fundamental functions; biorthogonal wavelets; dual functions; interpolatory subdivision scheme; refinable functions

Document Type: Original Article

Affiliations: 1: The National University of Singapore, Singapore, 2: University of Alberta, Edmonton, Alberta, Canada

Publication date: February 1, 1999

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