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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 GMRA to interval [0,1].

e build on results of the section ( Calculation of biorthogonal wavelets ) and use technique of the section ( Adapting MRA to the interval [0,1] ).


(Biorthogonal scaling functions) A pair of scaling functions $\phi,\tilde{\phi}$ comes from dual GMRA, is compactly supported and

1. the associated wavelets $\psi,\tilde{\psi}$ satisfy MATH for some $N,\tilde{N}$ : MATH

2. the supports of $\phi,\tilde{\phi}$ satisfy the condition MATH

Therefore, according to the proposition ( Reproduction of polynomials 4 ), MATH

According to the definition ( Dual wavelets ), MATH

We introduce the notations $K_{d}$ , $K_{d,L}$ , $K_{d,R}$ , $K_{d,I}$ , $\tilde {K}_{d}$ , $\tilde{K}_{d,L}$ , $\tilde{K}_{d,R}$ , $\tilde{K}_{d,I}$ similarly to the section ( Adapting MRA to the interval [0,1] ): MATH MATH According to the condition ( Biorthogonal scaling functions )-2, MATH


(Sufficiently fine scale 2) We assume that the parameter $d$ is sufficiently large so that MATH

For a polynomial MATH we have MATH We introduce the notation MATH In particular, MATH


([0,1]-adapted GMRAs) We define the spaces MATH MATH

We proceed to construct biorthogonal bases for MATH , MATH for each $d$ .

Let MATH MATH We choose MATH , MATH , MATH , MATH to satisfy MATH

We represent the relationships MATH and MATH as MATH MATH for some matrixes $A,\tilde{A}$ chosen to satisfy MATH : MATH so that MATH

If $N=\tilde{N}$ then $X$ is a square matrix of the form MATH with some square matrixes $X_{1},X_{2}.$ The LU decomposition may be applied separately to the matrixes $X_{1},X_{2}$ : MATH for some permutation matrixes $P_{i}$ , lower triangular matrixes $L_{i}$ and upper triangular matrixes $U_{i}$ . The choice MATH delivers one possible solution.

Notation. Index. Contents.

Copyright 2007