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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.
a. Adapting MRA to the interval [0,1].
b. Adapting wavelets to interval [0,1].
c. Adapting GMRA to interval [0,1].
d. Adapting dual wavelets to interval [0,1].
e. Constructing dual GMRA on [0,1] with boundary conditions.
3. Finite element method.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Adapting dual wavelets to interval [0,1].


e assume that the conditions ( Biorthogonal scaling functions ) and ( Sufficiently fine scale 2 ) hold and build on results of the previous section. We assume that $N=\tilde{N}$ and MATH .

We introduce the spaces MATH and MATH via the relationships MATH

The bases for MATH and MATH are constructed as follows. Let MATH The space MATH has dimension MATH . Hence, we construct a part of the basis for MATH by taking MATH and MATH . The remaining $2N$ functions from $W_{d}$ are derived by constructing the functions

MATH (Wavelets on 01 step 1)
where MATH Similarly, on the right hand side of the interval $\left[ 0,1\right] $ we construct the same $\zeta_{d,k_{1}}$ and MATH by setting MATH We form the linear combinations MATH MATH for some finite sequences MATH , MATH , MATH , MATH determined below. The wavelet dual bases of the space $W_{d}$ take the form MATH The MATH , MATH , MATH , MATH are chosen to satisfy MATH

Let MATH for some matrixes $A,\tilde{A}$ and columns MATH Then MATH

We conclude as in the previous section: MATH MATH MATH

Remark

(Dimension mismatch) Note that we generally have MATH This means that we get at least 2 functions too many when constructing basis for the space $W_{d}$ . As a result, when we perform the biorthogonalization of boundary functions, at least two of the resulting functions (one for every boundary) vanish, see the proposition ( Linear independence from biorthogonality ). We discard such functions and end up with a basis of correct dimensionality.





Notation. Index. Contents.


















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