3.1. Biorthogonal Multiwavelet Bases of
We construct a pair of biorthogonal multiwavelet bases of using the original multiwavelet systems , and the derived systems , of . Divergence-free wavelets satisfying vanishing normal boundary conditions can then be constructed through tensor products from a basis of generated from the smoothed multiwavelet system vanishing on the boundary of its support.
Our procedure for constructing biorthogonal multiwavelet bases of
adapted from those of
can be described as follows:
Keep the functions that are originally supported on ,
For the functions belonging to the original biorthogonal systems of whose support straddles the boundary point 0, truncate the symmetric ones to and normalize them by , and shift the antisymmetric ones to .
For the functions belonging to the smoothed and roughened systems of whose support straddles the boundary point 0, truncate the antisymmetric ones to and normalize them by , and shift the symmetric ones to .
Precisely, for
, we define
and
Use the same formulation for
and
. Let
and similarly for
in terms of
, respectively. We obtain a pair of biorthogonal MRAs
and
of
such that
and , just as in the case of the whole real line,
We perform a similar procedure for the construction of the biorthogonal multiwavelet bases of
generated from the new systems
and
of
except that the roles of symmetric and antisymmetric components whose support overlaps 0 are switched. Explicitly, for
, let
and
We define similarly for
and
. Let
and similarly for
in terms of
, respectively. We get another pair of biorthogonal MRAs
and
of
such that
and
To adapt the differentiation and integration relations between the scaling vectors and multiwavelets on
to ones on
, we separate the scaling vectors and multiwavelets on
into
boundary and
interior components. We define the
boundary scaling vectors and multiwavelets, which correspond to the integer translate
, as follows
We define
and
similarly to
and
, respectively. The
interior components, formulated as below, are the scaling vectors and multiwavelets on
with the integer translates
, which live completely inside
. Let
and similarly for
and
.
From Equations (10) and (14), and Equations (19) and (21) the multiwavelets on
inherit the same differentiation and integration relations as the multiwavelets on
. Precisely,
for both interior and boundary components
and
.
The boundary and interior scaling vectors on
are less straightforward. From Equations (9) and (10) and Equations (18) and (20) that define the scaling functions on
we obtain
for the respective boundary and interior scaling vectors, where
To establish an analogue of Proposition 2.1 on
, we define oblique projections
,
from
onto the respective approximation spaces
and
and
,
from
onto the corresponding detail spaces
and
as follows:
For an interval
, possibly unbounded, the Sobolev space
is the Hilbert space defined by
Define
where
is the space of continuously differentiable functions compactly supported in Ω. Note that if
then
if and only if
on
.
Proposition 3.1. On the Sobolev space ,
the following commutation relations hold The proof of Proposition 3.1 can be found in
Appendix B.
3.2. Construction of Divergence-Free Multiwavelets
The following are some basic notions of flux spaces and divergence-free vector fields. Denote
the upper half plane, and
The
divergence operator is defined as usual by
where the partial derivatives are understood in the distributional sense. The divergence operator induces the
flux space
and its
divergence-free subspace
The two spaces of vector fields are Hilbert spaces under the norm
where
We have found biorthogonal multiwavelet systems related by differentiation for both and so that the commutation relations between oblique projections and differentiation are all satisfied. We are now able to construct a wavelet basis for the vector space satisfying the vanishing normal boundary condition , where is the unit outward normal vector to the boundary axis .
We have utilized many notations so far. To avoid confusion, we recall the notations and relations that are necessary for the construction.
Biorthogonal scaling vectors and and wavelets and on , related by Equation (10).
Biorthogonal boundary scaling vectors and and wavelets and on , related by related by Equations (22) and (23).
Biorthogonal interior scaling vectors and and wavelets and , on , related by Equations (22) and (23). These multiwavelet systems establish the commutation relations as in Propositions 2.1 and 3.1.
Our construction of biorthogonal bases of compactly supported multiwavelets on such that the reconstruction wavelets are divergence-free will be divided into the following steps.
Step 1. Compose biorthogonal multiwavelet bases in by tensor products.
We use the standard basis vectors
and
to index a smoothing direction for tensor wavelets on
:
Similarly, we use negated standard basis vectors to index roughening directions for dual tensor scaling and wavelet spaces on
:
The decomposition corresponding to Equation (24) holds for the respective indices
and
.
We define the
boundary generators of
to be components of the matrices
Here
is biorthogonal to
, that is,
. We have another set of boundary biorthogonal generators of
given by
We define similarly biorthogonal
interior generators of
:
and
,
, where each boundary scaling vector or multiwavelet is replaced by the corresponding interior one. For instance,
Along with the boundary generators, they constitute two biorthogonal bases of
as listed below:
with the duals ;
with the duals ; .
Step 2. Compose biorthogonal bases of componentwise.
The biorthogonal bases of
induce biorthogonal bases of
componentwise. In fact,
with the dual components
form biorthogonal bases of
. The following is the list of the boundary matrix generators. The interior generators and their duals are formulated similarly using the appropriate substitution of the boundary vector by the interior vector. Explicitly,
Step 3. Compose the biorthogonal bases in .
We can obtain a biorthogonal basis in
from the linear combinations of the vector fields listed above and their integer translates. The following are the reconstruction boundary multiwavelets of the basis:
Because each component of
and of
vanishes continuously at
, these boundary multiwavelets satisfy the vanishing normal condition, that is, their normal component vanishes continuously at the boundary.
We formulate the reconstruction interior multiwavelets
similarly, where
and the boundary component are substituted respectively by
and the corresponding interior component. For instance,
The vectors
and
,
belong to the divergence-free vector space
because of the commutation relations between oblique projections and differentiation on both
and
specified by Propositions 2.1 and 3.1. We prove below that the
reconstruction wavelets and
,
constitute a basis for
. Their biorthogonal duals, which serve as the decomposition wavelets, are
for the boundary components. The interior components
are defined similarly. For instance
Notice that the decomposition multiwavelets are not divergence-free.
Theorem 3.1. For , the expansion of F in terms of agrees with its expansion in terms of . Thus, the translates and dilates of form a basis for the divergence-free subspace of whose boundary components satisfy the vanishing normal boundary condition.
Proof. It suffices to verify the boundary case. One shows that if a vector field lies in the divergence-free subspace, then its expansion in terms of a complete set of vector wavelets for agrees with its sum of its components in the divergence-free wavelets.
Let
. Its boundary expansion in terms of the biorthogonal bases
of
is represented by the six following vector fields and the components of their shifts and dilates:
On the other hand, the boundary expansion of
F in terms of the divergence-free wavelets
is represented by the following fields and the components of their translates and dilates:
Under the hypothesis that F is divergence-free, we show that . By definition, we have and . Using the differentiation relations between the scaling vectors and multiwavelets on and , and the commutation relations in Propositions 2.1 and 3.1, we show that and .
Since the
y-coordinates of
are equal to 0 for
, we can assume that
. Furthermore, all scaling and wavelet functions on
vanish on the boundary of their support. Thus,
This implies
. In addition,
The verification for is similar.
Each one of the divergence-free multiwavelets
consists of nine components.
Figure 4 plots some of the components of
.
Figure 4.
Some components of .
Figure 4.
Some components of .