Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 A. Elementary definitions of wavelet analysis.
 B. Haar functions.
 C. Multiresolution analysis.
 D. Orthonormal wavelet bases.
 E. Discrete wavelet transform.
 F. Construction of MRA from scaling filter or auxiliary function.
 G. Consequences and conditions for vanishing moments of wavelets.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 J. Construction of (G)MRA and wavelets on an interval.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Wavelet Analysis.

he present section is a necessary prerequisite for the section ( Sparse tensor product ). The reference is [Walnut] .

Wavelets are a preferable way to construct approximation spaces when applying finite element method to PDEs. Wavelets replicate polynomials and thus have efficiency of approximation. Wavelet decompositions have natural and stable subspace splittings and thus allow for preconditioners suitable for parallel calculations. Wavelets form bases suitable for sparse tensor product-based representation. Such bases grow conservatively when increasing dimensionality. Multiscale structure of wavelet bases is naturally suitable for construction of adaptive grids.

The idea of wavelets may be illustrated by considering an attempt to approximate generic functions with elementary shapes. We introduce a mesh and define a constant function on every interval : As we increase the scale we get finer approximation. Note that is obtained by scaling and translation from the elementary shape The closures of linear spans form and increasing sequence of spaces The wavelets arise when we decide not to discard information while going from to . Instead, we would like to produce a basis of the increment space : Such structure is essential when constructing adaptive grid.

We would like such basis to have the form span and span for some functions and . In addition, we may want to increase complexity of the elementary shape so that would include polynomials up to some degree. Moreover, we want supports of and to be finite and minimal. We also may want to have symmetry of some kind. Then we might want to restrict such construction to an interval and force it to satisfy boundary conditions. Finally, we need such construction to have strong stability with respect to subspace decompositions. It is remarkable that we can have it all. The following sections contain detailed derivations and occasional Mathematica scripts. The part ( Numerical analysis part ) contains some practical Python scripts and C++/Cuda codes.

 A. Elementary definitions of wavelet analysis.
 B. Haar functions.
 C. Multiresolution analysis.
 D. Orthonormal wavelet bases.
 E. Discrete wavelet transform.
 F. Construction of MRA from scaling filter or auxiliary function.
 G. Consequences and conditions for vanishing moments of wavelets.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 J. Construction of (G)MRA and wavelets on an interval.
 Notation. Index. Contents.