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 biortho gonal wavelet. In this paper, we shall investigate the mutual relations amo ng these three properties. A characterization of L-p (1 less than or equal to p less than or equal to infinity) smoothness of multivariate refinable f unctions is presented. It is well known that there is a close relation betw een a fundamental refinable function and a biorthogonal wavelet. We shall d emonstrate that any fundamental refinable function, whose mask is supported on [1 - 2r; 2r - 1](s) for some positive integer r and satisfies the sum r ules of optimal order 2r, has L-p smoothness not exceeding that of the univ ariate fundamental refinable function with the mask b(r). Here the sequence b(r) on Z is the unique univariate interpolatory refinement mask which is supported on [1 - 2r; 2r - 1] and satisfies the sum rules of order 2r. Base d on a similar idea, we shall prove that any orthogonal scaling function, w hose mask is supported on [0, 2r - 1](s) for some positive integer r and sa tisfies the sum rules of optimal order r, has L-p smoothness not exceeding that of the univariate Daubechies orthogonal scaling function whose mask is supported on [0; 2r 1]. We also demonstrate that a similar result holds tr ue for biorthogonal wavelets. Examples are provided to illustrate the gener al theory. Finally, a general CBC (cosets by cosets) algorithm is presented to construct all the dual refinement masks of any given interpolatory refi nement mask with the dual masks satisfying arbitrary order of sum rules. Th us, for any scaling function which is fundamental, this algorithm can be em ployed to generate a dual scaling function with arbitrary approximation ord er. This CBC algorithm can be easily implemented. As a particular applicati on of the general CBC algorithm, a TCBC (triangle cosets by cosets) algorit hm is proposed. For any positive integer k and any interpolatory refinement mask a such that a is symmetric about all the coordinate axes, such a TCBC algorithm provides us with a dual mask of a such that the dual mask satisf ies the sum rules of order 2k and is also symmetric about all the coordinat e axes. As an application of this TCBC algorithm, a family of optimal bivar iate biorthogonal wavelets is presented with the scaling function being a s pline function.