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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.

Consequences and conditions for vanishing moments of wavelets.


f the wavelet $\psi$ is vanishing quickly enough and is sufficiently smooth then it has vanishing moments MATH and the linear span of MATH contains polynomials up to some degree. These lead to good approximation properties of the subspaces MATH . The following subsections spell out such statements.




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.

Notation. Index. Contents.


















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