## Abstract

In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavelets, in this paper, we shall discuss the mutual relations among these three properties. For example, we shall see that any orthogonal scaling function, which is supported on $[0,2r-1]^s$ for some positive integer $r$ and has accuracy order $r$, has $L_p$ $(1\le p \le \infty)$ smoothness not exceeding that of the univariate Daubechies orthogonal scaling function which is supported on $[0,2r-1]$. Similar results hold true for fundamental refinable functions and biorthogonal wavelets. Then, we shall discuss the relation between symmetry and the smoothness of a refinable function. Next, we discuss the construction by cosets (CBC) algorithm reported in Han \cite{\Hanbw} to construct biorthogonal wavelets with arbitrary order of vanishing moments. We shall generalize this CBC algorithm to construct bivariate biorthogonal wavelets. For any positive integer $k$ and a bivariate primal mask $a$ such that $a$ is symmetric about the origin, such CBC algorithm provides us a dual mask of $a$ such that the dual mask satisfies the sum rules of order $2k$ and is also symmetric about the origin. The resulting dual masks have certain optimal properties with respect to their support. Finally, examples of bivariate biorthogonal wavelets constructed by the CBC algorithm are provided to illustrate the general theory. Advantages of the CBC algorithm in this paper over other methods on constructing biorthogonal wavelets are also discussed.

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