e build on results of the
section (
Calculation
of biorthogonal wavelets
) and use technique of the section
(
Adapting MRA to the
interval [0,1]
).
Therefore, according to the proposition
(
Reproduction of polynomials
4
),
According to the definition (
Dual
wavelets
),
We introduce the notations
,
,
,
,
,
,
,
similarly to the section
(
Adapting MRA to the
interval
[0,1]
):
According to the condition
(
Biorthogonal scaling
functions
)2,
Condition
(Sufficiently fine scale 2) We assume
that the parameter
is sufficiently large so
that
For a polynomial
we
have
We introduce the
notation
In
particular,
Definition
([0,1]adapted GMRAs) We define the
spaces
We proceed to construct biorthogonal bases for
,
for each
.
Let
We choose
,
,
,
to
satisfy
We represent the relationships
and
as
for some matrixes
chosen to satisfy
:
so
that
If
then
is a square matrix of the
form
with some square matrixes
The LU decomposition may be applied separately to the matrixes
:
for some permutation matrixes
,
lower triangular matrixes
and upper triangular matrixes
.
The
choice
delivers one possible solution.
