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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
a. Scaling equation.
b. Support of scaling function.
c. Piecewise linear MRA, part 1.
d. Orthonormal system of translates.
e. Approximation by system of translates.
f. Orthogonalization of system of translates.
g. Piecewise linear MRA, part 2.
h. Construction of MRA summary.
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.

Orthogonalization of system of translates.


(OST property 3) Assume that

1. MATH ,

2. $g$ has compact support,

3. there exist constants $A,B>0$ such that

MATH (Riesz basis condition)
then there exists a function $g_{0}$ such that


b. MATH is OST,

c. MATH . The closure is taken in $L^{2}$ .


We look for $g_{0}$ in MATH . Then (a) is trivial and (c) would be satisfied (see below). To get (b) we use the proposition ( OST property 1 ).

We set MATH We need MATH to achieve (a) because MATH (see the formula ( Property of scale and transport 3 )) and we need MATH to achieve (b) (see the proposition ( OST property 1 )). We take the Fourier transform of $\left( \#\right) $ : MATH We use the formula ( Property of scale and transport 4 ): MATH Hence MATH Therefore, we would like to choose MATH to satisfy $\left( \&\right) $ according to the rule MATH Hence, we have MATH if we establish that the RHS is in MATH (see the sections ( Fourier series ) and ( Fourier analysis in Hilbert space )). But this is evident from the conditions (1),(2) and (3).

It remains to prove that $\left( \#\right) $ with MATH implies (c). The statement MATH is trivial. We need to prove MATH . Let MATH so that MATH We seek MATH such that MATH or MATH

Thus we need to have MATH and MATH under assumptions that MATH . Furthermore, by the condition (2), the summation in $p$ and the number of MATH is finite. Hence, MATH is finite dimensional linear equation. The matrix of MATH has the form MATH for some index ranges $K$ and $P$ of the same length depending on diameter of $spt~g$ . Since MATH has only a finite number non-zero coordinates, the matrix $C$ is semi-diagonal. Hence, $\det C\not =0$ and MATH always have a solution.

Notation. Index. Contents.

Copyright 2007