dapting scaling
functions to the interval [0,1] is based on the technique of the section
(
Adapting dual
wavelets to interval [0,1]
) and remarks of the previous section. The
python code is located in the file
"OTSProjects\python\wavelet\interval\interval.py".
The classes PsiGenerator and PsiGenerator2 implement the procedure. The script
"_run_interval.py" performs the calculations.
We modify the procedure to reflect the fact that the functions
are calculated with significantly better precision than the functions
.
We replace the formulas (
Wavelets on 01 step
1
) with the following procedure that does not involve
.
Note that the formulas (
Wavelets on 01 step
1
) are of the form
where the
and
are biorthogonal bases. The goal is to do without
.
Let
be the orthogonal projection of
on the linear span of the finite set
.
Thus
for some numbers
.
We apply the operation
and
obtain
In matrix
notation,
If the set
is linearly independent then the matrix
is nondegenerate. In
addition,
where
Thus
and we recover the
for the
formula
by solving the
system
After we recover the boundary functions
we do not perform biorthogonalization because we cannot (the
have poor precision) but also because there is no need. Indeed, to conduct
finite element calculations we need linear independence, approximation
properties and subspace decomposition stability. But then, in light of the
remark (
Dimension mismatch
), we need to
separate correct number of basis functions. This is the motivation for the
following sections.
