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.

## Calculation of scaling functions.

he next step is to implement the cascade algorithm (see the section ( Recovering scaling function from auxiliary function )). The cascade algorithm starts from a finite support function and then proceeds The function may be chosen as However, the procedure is such that if is piecewise constant then is piecewise constant. Other piecewise polynomial expressions are acceptable as long as the support remains finite and For this reason we explore several possibilities for to a be a spline function (see the section ( Spline functions )) of increasing complexity.

The absolutely minimal degree of smoothness is . Such requirement will be evident when we consider applications to finite element technique.

The cascade procedure is implemented in the function "cascade" of the class "FourFunctions" in the file "OTSProjects\python\wavelet\cascade.py".