Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function phi which has sufficient regularity and which is fundamental for interpolation [that means, phi(0)=1 and phi(a lpha)=0 for all alpha is an element of Z(s)\{0}]. Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. Thi s article presents several schemes to construct compactly supported fundame ntal refinable functions, which have higher regularity, directly from a giv en, continuous, compactly supported, refinable fundamental function phi. As ymptotic regularity analyses of the functions generated by the construction s are given. The constructions provide the basis for multivariate interpola tory 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 bio rthogonal wavelets. Combined with the biorthogonal wavelet construction alg orithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets w hich have arbitrarily high regularity. Several examples are computed.