### Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

### Wolfgang Dahmen, Bin Han, Rong-Qing Jia and Angela Kunoth

## Abstract

Starting with Hermite cubic splines as primal multigenerator,
first a dual multigenerator on $\R$ is constructed which
consists of continuous functions, has small support and
is exact of order two. We then derive multiresolution
sequences on the interval while retaining the polynomial
exactness on the primal and dual side. This guarantees
moment conditions of the corresponding wavelets.
The concept of stable completions
\cite{CDP} is then used to construct corresponding primal
and dual multiwavelets on the interval as follows.
An appropriate variation of what is known as hierarchical
basis in finite element methods is shown to be an
initial completion. This is then in a second step
projected into the desired complements spanned by
compactly supported biorthogonal multiwavelets.
The masks of all multigenerators and multiwavelets
are finite so that decomposition and reconstruction algorithms
are simple and efficient.
Furthermore, in addition to Jackson estimates which follow
from the exactness, one can also show Bernstein inequalities
for the primal and dual multiresolution.
Consequently, sequence norms for the coefficients based on
such multiwavelet expansions characterize Sobolev norms
$\|\cdot\|_{H^s([0,1])}$ for $s\in (-0.824926,2.5)$.
In particular, the multiwavelets form Riesz bases for
$L_2([0,1])$.

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