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] ).

Condition

(Biorthogonal scaling functions) A pair of scaling functions comes from dual GMRA, is compactly supported and

1. the associated wavelets satisfy for some :

2. the supports of satisfy the condition

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

According to the definition ( Dual wavelets ),

We introduce the notations , , , , , , , similarly to the section ( Adapting MRA to the interval [0,1] ): According to the condition ( Biorthogonal scaling functions )-2,

Condition

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

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

Definition

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

We proceed to construct biorthogonal bases for , for each .

Let We choose , , , to satisfy

We represent the relationships and as for some matrixes chosen to satisfy : so that

If then is a square matrix of the form with some square matrixes The LU decomposition may be applied separately to the matrixes : for some permutation matrixes , lower triangular matrixes and upper triangular matrixes . The choice delivers one possible solution.

 Notation. Index. Contents.