### Multiwavelets on the Interval

### Bin Han and Qing-Tang Jiang

## Abstract

Smooth orthogonal and biorthogonal multiwavelets on the real line with
their scaling function vectors being supported on $[-1,1]$ are
of interest in constructing wavelet bases on the interval $[0,1]$ due to their
simple structure. In this paper, we shall present a symmetric $C^2$ orthogonal
multiwavelet with multiplicity $4$ such that its orthogonal scaling function vector
is supported on $[-1,1]$, has
accuracy order $4$ and belongs to the Sobolev space $W^{2.56288}$.
Biorthogonal multiwavelets with multiplicity $4$ and vanishing
moments of order $4$ are also constructed such that the primal scaling function
vector is supported on $[-1,1]$,
has the Hermite interpolation properties and belongs to $W^{3.63298}$
while the dual scaling function vector is supported on $[-1,1]$ and
belongs to $W^{1.75833}$.
A continuous dual scaling function vector of the cardinal Hermite interpolant
with multiplicity $4$ and support $[-1,1]$ is also given.
Based on the above constructed orthogonal and biorthogonal
multiwavelets on the real line,
both orthogonal and biorthogonal multiwavelet bases on
the interval $[0,1]$ are presented.
Such multiwavelet bases on the interval $[0,1]$ have
symmetry, small support, high vanishing moments, good smoothness
and simple structures. Furthermore,
the sequence norms for the coefficients based on such
orthogonal and biorthogonal multiwavelet expansions characterize Sobolev
norm $\|.\|_{W^s([0, 1])}$ for $s\in (-2.56288, 2.56288)$
and for $s\in (-1.75833, 3.63298)$, respectively.

Back to Preprints and publication