Analysis and Construction of Optimal Multivariate Biorthogonal Wavelets with Compact Support

In applications, it is well known that high smoothness, small support, and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of $L_p\, (1\le p \le \infty)$ smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1-2r,2r-1]s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br. Here the sequence br on ${\Bbb Z}$ is the unique univariate interpolatory refinement mask which is supported on [1-2r,2r-1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal scaling function, whose mask is ...