Quantitative 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.
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.

Elementary definitions of wavelet analysis.


e already introduced the mesh MATH in the previous section.

Given MATH , the index $k$ is called the "location index" and the index $d$ is called the "scale" index.

Definition

(Scale and transport operators) We introduce the operations MATH and MATH according to the rules MATH

Note that MATH MATH We introduce the notations $\Delta$ , $\QTR{cal}{T}^{d}$ , $\Delta_{dk}$ , $~h_{d}$ , $\Delta_{dk}^{\pm}$ : MATH

Definition

(Scale and transport operators 2) We introduce the operations MATH and MATH according to the rules: MATH

Note that MATH MATH Read the above " $x\in$ spt $~S_{d}^{\ast}u$ (see MATH ) iff $S_{-d}x\in$ spt $~u$ (see MATH )". Consequently MATH Similarly, MATH We conclude

MATH (Property of scale and transport 1)
In addition,
MATH (Property of scale and transport 2)
MATH (Property of scale and transport 3)

In the following sections we frequently use Fourier transform $^{\wedge}$ as well as the operations $T^{\ast},S^{\ast}$ . We investigate the interaction:

MATH (Property of scale and transport 4)

MATH We make a change $y=2^{d}x$ ,

MATH (Property of scale and transport 5)

MATH (Property of scale and transport 6)

We investigate transposition of $S_{d}^{\ast}$ and $T_{k}^{\ast}$ :

MATH (Property of scale and transport 7)

Definition

(Orthonormal wavelet basis) The set of functions of the form MATH , MATH is called "orthonormal wavelet basis" if it is orthonormal: MATH

and constitutes a basis in MATH .





Notation. Index. Contents.


















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