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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 wavelets to interval [0,1].


e assume that the conditions ( Compactly supported scaling function ) and ( Sufficiently fine scale ) hold and build on results of the previous section.

Notation

We introduce the notation MATH where the orthogonal complement is taken with respect to the MATH -scalar product.

The above is to be compared with the notation ( Notation W_d ) and the proposition ( Existence of orthonormal wavelet bases 1 ).

We proceed to construct an MATH -orthogonal basis for MATH .

Proposition

(Existence of adapted wavelets) For each $d$ we define the functions MATH : MATH Then

1. MATH are linearly independent,

2. MATH ,

3. MATH are MATH -orthogonal to all MATH .

Proof

The claim 1 follows by contradiction. The collection MATH is a part of a finite basis for MATH . The dimension of MATH increases by factor of $2$ when increasing $d$ by $1$ . Linear dependence of MATH would mean that MATH (and thus MATH ) is missing a dimension. This contradicts the proposition ( Maximal dimension 1 ).

According to the formula $\left( \#\right) $ , MATH According to the construction of $\eta_{d,k}^{L}$ , the functions MATH are orthogonal to MATH for every $d$ . Hence, MATH This is the claim 2.

According to the proposition ( Scaling equation 2 ), MATH and according to the proposition ( Support of scaling function ) and the condition ( Compactly supported scaling function ), MATH MATH MATH and, as already noted, MATH are orthogonal to all MATH . Hence, we have the claim 3.

It remains to perform orthogonalization of MATH similarly to the orthogonalization of $x_{d,L}^{m}$ leading to $\eta _{d,k}^{L}$ in the previous section.





Notation. Index. Contents.


















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