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Analysis and construction of optimal
multivariate biorthogonal wavelets with compact support

### Bin Han

## Abstract

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 $L_p$ smoothness not
exceeding that of the univariate
fundamental refinable function with the mask $b_r$. Here the sequence
$b_r$ on $\bZ$ 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
supported on $[0,2r-1]^s$ for some positive integer $r$
and satisfies 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 true for biorthogonal
wavelets. Examples are provided to illustrate the general theory.
Finally, a general CBC (Construction By Cosets) algorithm is presented
to construct all the dual
refinement masks of any given interpolatory refinement mask with the
dual masks satisfying arbitrary order of sum rules. Thus, for any
scaling function which is fundamental, this algorithm can be employed
to generate a dual scaling function with arbitrary approximation order.
This CBC algorithm can be easily implemented.
As a particular application of the general CBC algorithm, a TCBC
(Triangle Construction By Cosets) algorithm is proposed. For any
positive integer $k$ and any interpolatory refinement mask $a$ such
that $a$ is symmetric about all the coordinate axes, such TCBC
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 all
the coordinate axes.
As an application of this TCBC algorithm,
a family of optimal bivariate biorthogonal wavelets is
presented with the scaling function being a spline function.

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