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.

## Convergence of modified cascade procedure.

n this section we use the crudification operator, see the definition ( Crudification operator ), to implement the properties ( Wavelet approximation 2 )-( Wavelet approximation 4 ) with close to machine precision and wavelet representations of reasonable size. We look at -based scaling function basis .

The script wavelet2\_run_cascade.py performs the following procedure.

Algorithm

(Modified cascade procedure) Choose a positive integer . For a given scaling filter (see the section ( Symmetric biorthogonal wavelets )) do the following.

1. Set Thus and a piecewise quadratic polynomial with finite support centered around : .

2. Calculate

3. Set and calculate Stop when numerical errors start to compete with convergence. Set

4. Calculate

Similarly calculate from the scaling filter .

For the PiecewisePoly library (see the section ( Piecewise polynomials in parallel )) compiled over double precision and calculated for we obtain

The script wavelet2\approxTest.py calculates the .

The same procedure is implemented in the scripts _l_run_cascade.py and l_approxTest.py using the class LPoly introduced in the section ( Manipulation of localized piecewise polynomial functions ). The convergence results are as follows. Thus, there is a significant improvement in approximation for high . The results are as follows. Thus, improvement in polynomial approximation is relatively slight.