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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.
a. Recursive relationships for wavelet transform.
b. Properties of sequences h and g.
c. Approximation and detail operators.
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.

Recursive relationships for wavelet transform.


roposition

(Recursive relationships for wavelet transform) We assume the setup ( Discrete wavelet transform setup ). For a function MATH we define MATH Then

a. MATH ,

b. MATH ,

c. MATH .

Proof

of (a). According to the proposition ( Scaling equation ), MATH We need to extend this relationship to any scale and location. MATH We use the formula ( Property of scale and transport 7 ). MATH Thus MATH Therefore, MATH

Proof

of (b). The wavelet $\psi$ is given by the equation MATH thus MATH We use the formula ( Property of scale and transport 7 ). MATH Therefore, MATH We calculate MATH

Proof

of (c). According to the definition ( Approximation and detail operators ) and propositions ( Existence of orthonormal wavelet bases 1 ), ( Existence of orthonormal wavelet bases 2 ) MATH MATH We put $s:=-s-1$ into $\left( \#\right) $ MATH and substitute the terms MATH We substitute $\left( \&\right) $ and MATH . MATH Thus MATH





Notation. Index. Contents.


















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