I. Wavelet calculations.
 1 Calculation of scaling filters.
 2 Calculation of scaling functions.
 3 Calculation of wavelets.
 5 Direct verification of wavelet properties.
 6 Adapting scaling function to the interval [0,1].
 7 Adapting wavelets to the interval [0,1].
 II. Calculation of approximation spaces in one dimension.
 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.

## Adapting wavelets to the interval [0,1].

dapting scaling functions to the interval [0,1] is based on the technique of the section ( Adapting dual wavelets to interval [0,1] ) and remarks of the previous section. The python code is located in the file "OTSProjects\python\wavelet\interval\interval.py".

The classes PsiGenerator and PsiGenerator2 implement the procedure. The script "_run_interval.py" performs the calculations.

We modify the procedure to reflect the fact that the functions are calculated with significantly better precision than the functions . We replace the formulas ( Wavelets on 01 step 1 ) with the following procedure that does not involve .

Note that the formulas ( Wavelets on 01 step 1 ) are of the form where the and are biorthogonal bases. The goal is to do without . Let be the orthogonal projection of on the linear span of the finite set . Thus for some numbers . We apply the operation and obtain In matrix notation, If the set is linearly independent then the matrix is non-degenerate. In addition, where Thus and we recover the for the formula by solving the system

After we recover the boundary functions we do not perform biorthogonalization because we cannot (the have poor precision) but also because there is no need. Indeed, to conduct finite element calculations we need linear independence, approximation properties and subspace decomposition stability. But then, in light of the remark ( Dimension mismatch ), we need to separate correct number of basis functions. This is the motivation for the following sections.