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Approximation Properties and Construction of Hermite
Interpolants and Biorthogonal Multiwavelets

### Bin Han

## Abstract

Multiwavelets are generated from refinable function vectors by using
multiresolution analysis. In this paper we investigate the
approximation properties of a multivariate refinable function vector
associated with a general dilation matrix in terms of
both the
subdivision operator and the order of sum rules satisfied by the
matrix refinement mask.
Based on a fact about
the sum rules of biorthogonal multiwavelets, a
coset by coset (CBC) algorithm is presented to construct
biorthogonal multiwavelets with arbitrary order of vanishing moments.
More precisely, to obtain biorthogonal
multiwavelets, we have to construct
primal and dual masks. Given any primal matrix mask $a$ and a general
dilation matrix $M$,
the proposed CBC algorithm
reduces the construction of all dual masks of $a$, which
satisfy the
sum rules of arbitrary order, to a problem of solving
a well organized system
of linear equations.
We prove in a constructive way that for any given
primal mask $a$ with a dilation matrix $M$ and for
any positive integer $k$, we can always construct a dual mask $\wt a$
of $a$ such that $\wt a$ satisfies the sum rules of order $k$.
In addition, we provide a general way for the construction of
Hermite interpolatory matrix masks in the univariate setting with any
dilation factors.
From such Hermite interpolatory masks, smooth Hermite interpolants,
including the well known cubic Hermite splines as a special case,
are obtained and are used to construct biorthogonal multiwavelets.
As an example, a $C^3$ Hermite interpolant with
support $[-3,3]$ is presented.
Then we shall apply the CBC algorithm to
such Hermite interpolatory masks to construct biorthogonal
multiwavelets.
Several examples of biorthogonal multiwavelets
are provided to illustrate the general theory.
In particular, a $C^1$ dual function vector with support
$[-4,4]$ of the cubic Hermite
splines is given.

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