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 where the orthogonal complement is taken with respect to the -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 -orthogonal basis for .

Proposition

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

1. are linearly independent,

2. ,

3. are -orthogonal to all .

Proof

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

According to the formula , According to the construction of , the functions are orthogonal to for every . Hence, This is the claim 2.

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

It remains to perform orthogonalization of similarly to the orthogonalization of leading to in the previous section.

 Notation. Index. Contents.