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 MRA to the interval [0,1].

verywhere in this section we assume that the following condition holds. See the proposition ( Existence of smooth compactly supported wavelets ) for context.

Condition

(Compactly supported scaling function) A scaling function comes from an MRA and is compactly supported: Moreover, the associated wavelet satisfies

Therefore, according to the proposition ( Reproduction of polynomials 4 ), where the is the collection of all polynomials with degree up to and including .

We introduce the notations so that

Condition

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

For a polynomial we introduce the notation In particular, and, according to the proposition ( Reproduction of polynomials 4 ),

Definition

([0,1]-adapted MRA) We define the spaces where the sign refers to the -orthogonality.

Proposition

The spaces , and are mutually -orthogonal.

Proof

The statement follows from construction: the collection is -orthogonal and the spaces , and are linear span of for different values of the index .

We would like to find an -orthogonal basis for the space . The functions are already -orthogonal. Before we perform Gram-Schmidt orthogonalization of and we need to establish that both collections are linearly independent.

Proposition

(Restricted linear independence) The collection is linearly independent on .

Proof

The proof may be found in [Lemarie] or [Meyer] . For a particular this may be verified directly.

Proposition

(Maximal dimension 1) The collection is linearly independent. The collection is linearly independent.

Proof

We introduce the notation We aim to establish that the collection is linearly independent then the statement would follow from the proposition ( Restricted linear independence ).

We have We make the change , , . where and, according to the proposition ( Integral of scaling function ), Thus and linear independence of follows from linear independence of .

We proceed with orthogonalization. We seek the functions such that We substitute the first relationship into the second: The is a square symmetric positive-definite matrix. Hence, there exists a Choleski decomposition: and the choice is sufficient to produce -orthogonal collection . The collection is -orthogonal to because it is a linear combination of .

Proposition

(Resolution structure for adapted scale functions) The spaces have the structure

Proof

We saw in the proof of the proposition ( Maximal dimension 1 ) that there exist two-way linear transformations where

Due to the conditions is suffices to show that may be represented as a linear combination of .

We calculate We use the proposition ( Scaling equation ). We use the formula ( Property of scale and transport 7 ). Here , , hence the index varies over . We intend to make a change , . Thus can only be even. We separate even and odd values of . For even we have , , hence, the becomes : For odd we have , , : We make a change , in the first sum and in the second sum and put these sums together because then would vary over : We have We continue At this point we invoke the proposition ( Sufficient conditions for vanishing moments )-d: We separate even and odd values of : Note that and thus the proposition ( Sufficient conditions for vanishing moments )-d applies for all values of and We continue The expression is within the linear combination of .

 Notation. Index. Contents.