Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
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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.

Haar functions.

aar functions are a simple motivational example for developments of this chapter.

Let MATH We introduce MATH We have MATH


(Main property of Haar functions 1) The set MATH (set of Haar functions) is a complete orthonormal system of functions in MATH . The set MATH spans the class $V^{d}$ for every $d=1,2,...$


The fact that MATH is an orthonormal system is a direct verification.

Observe that any function $u\in V^{d}$ is defined by set of $2^{d}$ values $u|_{\Delta_{dk}}$ , $k=1,...,2^{d}$ and the set MATH is linearly independent (because it is orthonormal) and has MATH members. Hence, MATH is a basis for $V^{d}$ .

It remains to note that MATH is dense in MATH .

Notation. Index. Contents.

Copyright 2007