Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities

I. Wavelet calculations.
1. Calculation of scaling filters.
2. Calculation of scaling functions.
3. Calculation of wavelets.
4. Convergence of cascade procedure.
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.
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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 MATH and then proceeds MATH MATH MATH The function $\eta_{0}$ may be chosen as MATH However, the procedure MATH is such that if $\eta_{0}$ is piecewise constant then $\eta_{m},\forall m$ is piecewise constant. Other piecewise polynomial expressions are acceptable as long as the support remains finite and MATH For this reason we explore several possibilities for $\eta_{0}$ to a be a spline function (see the section ( Spline functions )) of increasing complexity.

The absolutely minimal degree of smoothness is $C^{1}$ . 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".

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Copyright 2007