I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 2 Convergence of modified cascade procedure.
 3 Calculation of boundary scaling functions II.
 IV. One dimensional problems.
 V. Stochastic optimization in one dimension.
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Calculation of boundary scaling functions II.

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