Multivariate Compactly Supported Fundamental Reenable Functions, Duals and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported reenable function with suucient regularity which is fundamental for interpolation (that means, (0) = 1 and () = 0 for all 2 Z s nf0g). Low regularity examples of such functions have been obtained numerically by several authors and a more general numerical scheme was given in RiS1]. This paper presents several schemes to construct compactly supported fundamental reenable functions with higher regularity directly from a given continuous compactly supported reenable 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 reenable functions, a necessary element in the construction of biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual reenable functions given in RiS2], we are thus able to obtain symmetric compactly supported multivariate biorthogonal wavelets with arbitrarily high regularity. Several examples are computed.