I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 1 Calculation of boundary scaling functions.
 2 Calculation of boundary wavelets.
 3 Testing properties of boundary wavelets and scaling functions.
 III. Calculation of approximation spaces in one dimension 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 wavelets.

e continue derivations of the previous section ( Calculation of boundary scaling functions ). We modify the procedure of the section ( Adapting dual wavelets to interval [0,1] ) to improve numerical stability.

Summary

(Calculation of boundary wavelets) Let be an basis for :

1. We calculate where

The projection is taken in .

2. We calculate where 3. We calculate for and .

4. We perform the -Gram-Schmidt orthogonalization separately for and . We denote the result .

In light of the remark ( Dimension mismatch ), some of the may have nearly zero -norm. We discard such functions and obtain the basis to match the requirement Afterwards, we -normalize the obtained functions.

The procedure is implemented in the script "bases\psi.py" and tested by the script "_run_bases.py". The flat version (no classes) of the procedure may be found in the file tApprox.py. The same file performs various tests. In particular, we verify directly that we have the scaling relationship and low condition numbers for various Gram matrixes.