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 scaling functions. otivation for calculations of this section was presented in the previous section ( Calculation of approximation spaces ).

We start from the pair introduced in the section ( Symmetric biorthogonal wavelets ) and calculated in the section ( Calculation of scaling functions ). We intend to adapt the procedure of the section ( Adapting GMRA to interval [0,1] ) to have two properties:

1. The procedure should be numerically stable.

2. The Gram matrix should have a low condition number.

The following is the procedure that accomplishes these two goals.

Summary

(Calculation of boundary scaling functions). Let and be the scaling filters calculated as described in the section ( Symmetric biorthogonal wavelets ). Let and be the output of the finite cascade procedure as described in the section ( Calculation of scaling functions ). Fix the scale parameter .

1. Form the sets as defined in the section ( Adapting GMRA to interval [0,1] ).

2. Calculate and recursively:  where the projection is taken in .

3. Calculate and : 4. Form the collection : 5. Calculate and recursively:   6. Form the basis of : The procedure is implemented in the script "bases\phi.py" and tested by the script "_run_bases.py". Experimentation shows that for 6 steps of scaling procedure we produce Gram matrix with condition number below 10 and precision of scaling projection below 0.0001.