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Quantitative Analysis
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Numerical Analysis
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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.
a. Vanishing moments vs decay at infinity.
b. Vanishing moments vs approximation.
c. Sufficient conditions for vanishing moments.
d. Reproduction of polynomials.
e. Smoothness of compactly supported wavelets with vanishing moments.
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.

Vanishing moments vs decay at infinity.


roposition

(Vanishing moments vs decay at infinity) Suppose that MATH and the collection MATH is $L^{2}$ -orthogonal with respect to the scale index: MATH

For any $N=0,1,...$ if MATH and MATH then MATH , $k=0,1,...,N$ .

Proof

We provide a proof by induction in $N$ .

In the case $N=0$ we have for MATH : MATH We use the proposition ( Basic properties of Fourier transform )-3. MATH We use the formula ( Property of scale and transport 6 ). MATH Hence, MATH We set MATH and choose MATH so that MATH for some $x_{0}$ to be determined later. By the inclusion MATH and the proposition ( Basic properties of Fourier transform )-5 MATH By the inclusion MATH and the proposition ( Dominated convergence theorem ), MATH We choose $x_{0}$ so that MATH then MATH

For general $N$ , we assume that the statement is already proven for $N-1$ . We have MATH By inclusion MATH we have MATH thus MATH We continue MATH MATH The integral in the above estimate for $I$ is bounded as MATH by inclusion MATH . By induction assumption MATH , $k=0,...,N-1$ . Hence, we cancel $\frac{1}{2^{dN}}$ and pass MATH to obtain MATH





Notation. Index. Contents.


















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