he procedures of summaries
(
Calculation of boundary
scaling functions
) and
(
Calculation of boundary
wavelets
) are reimplemented using Cuda/C++ based polynomial library within
the script wavelet2\basis2.py. Scaling functions of the previous section were
used as a starting point.
The same scripts tests polynomial approximation and scaling relationships. The
polynomial approximation test returns very good results. Scaling relationships
are poor, slightly improving with rising
.
The
-approximation
is especially difficult.
Because of such properties, it is more efficient to use high
to calculate boundary scaling functions, then apply crudification operator to
the resulting wavelet basis and abandon scaling equation. This is so because
scaling equation is not frequently applicable when calculating sparse tensor
product based decompositions.
The source of poor scaling properties is the step 3 of the procedure
(
Calculation of boundary
scaling functions
). The function
is calculated with significantly worse precision, as shown in the previous
section. The author experimented with a possibility to remove
from the procedure using the fact
and compactness of support of
.
Indeed,
must be a finite linear combination of
.
It is possible to significantly improve
-approximation
of scaling equation using this observation. However,
-approximation
of scaling equation does not show significant improvement. The author did not
pursue it further.
We remove
from the step 3 of the procedure
(
Calculation of boundary
scaling functions
) as follows.
According to the proposition
(
Reproduction of polynomials 4
),
for a polynomial
of degree below
we
have
for some numbers
and non-orthogonal family
.
We seek to replace
with
:
for some integer
.
As noted above, we expect existence of some finite
s.t.
.
In practical application, there are numerical errors increasing with
.
We
define
where the projection is taken in
.
Then we
set
and
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