dapting
scaling functions to the interval [0,1] is based on the technique of the
section (
Adapting GMRA to
interval [0,1]
). The python code is located in the file
"OTSProjects\python\wavelet\interval\interval.py".
The class PhiGenerator performs the task. The notation in the code follows the
notation of the section
(
Adapting GMRA to interval
[0,1]
). The example of use may be found in the function "makePhiGen" in
the file "_run_interval.py". The same file performs various direct
verifications of correctness of the procedure.
Examination of the code in the file interval.py reveals that we slightly
deviate from the procedure of the section
(
Adapting GMRA to interval
[0,1]
). The reason is numerical stability. The procedure is based on the
scaling functions
,
that come from infinite cascade procedure. We have to stop the cascade
procedure at some limited number of steps. As a result, we do not get exact
biorthogonality of the translation bases around
.
The biorthogonality is essential for the procedure of the section
(
Adapting GMRA to interval
[0,1]
). Without it we do not get biorthogonality between internal and
boundary scaling functions. If we start with a slight deviation from
biorthogonality of
then the error is amplified when we multiply by the matrixes
and
.
To see this, calculate eigenvalues for the matrixes
and
.
For symmetric biorthogonal scaling functions, calculated directly from the
procedure of the section
(
Adapting GMRA to interval
[0,1]
) and
(the smallest, simplest case), the 5 eigenvalues range from 2.5e1 to 2.0e11.
This is the origin of numerical troubles that follow after that. On the other
hand, achieving biorthogonality of
with great precision is costly.
The difficulty originates in the approximation properties of the system
that we use when constructing the initial boundary functions
,
,
,
.
As we increase the maximal degree, the condition number of Gram matrix
increases. To remove the difficulty, we rotate the system
into shifted Legendre polynomials. Indeed, the matrix
has eigenvalues in the range from 1.5 to 3.3e6 for
(see the script tPwrOrt.py) but the same matrix for the Legendre polynomials
is equal to
.
The more efficient solution is obtained by performing orthogonalization of
polynomials
with custom scalar
product:
for some positive on
function
.
Indeed, our concern is preserving biorthogonality of
vs
(biorthogonality across the boundary index
,
and similarly on the right side). We expect one of the quantities
,
,
,
to deviate the most from zero. Therefore, we obtain good stability if we
select
from
.
Experimentation shows (see the classes CondNumberMinimizer and
PolynomialMinimizer and the script _run_interval.py) that optimization of
does not provide significant improvement over such recipe.
In addition, we use the Python module "decimal" to control numerical
precision.
With these two modifications in place, it is possible to achieve
precision
for 8 steps of cascading procedure. We note for future use that only the
functions
are affected by the numerical instability because these are obtained via
multiplication of
with largevalued matrixes
.
The functions
are obtained via multiplication with
and, thus, are calculate with significantly better precision.
Our final remark is about extending results for scaling parameter
to
.
Note that
and
may be used interchangeably to construct MRA (GMRA for pair). The spaces
would be the same after shifting the parameter
.
Hence, we can recover boundary functions for
by scaling and shifting the
scale
boundary functions. We obtain the boundary functions
by applying the operation
to the functions
for some translation parameter
.
The
is selected by comparing
supports:
We perform a similar operation on the
right:
