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 OE with sufficient regularity which is fundamental for interpolation (that means, OE(0) = 1 and OE(ff) = 0 for all ff 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 refinable functions with higher regularity directly from a given continuous compactly supported refinable fundamental function OE. 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 the construction of biorthogonal wav..