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.
 a. Recursive relationships for wavelet transform.
 b. Properties of sequences h and g.
 c. Approximation and detail operators.
 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.

## Recursive relationships for wavelet transform.

roposition

(Recursive relationships for wavelet transform) We assume the setup ( Discrete wavelet transform setup ). For a function we define Then

a. ,

b. ,

c. .

Proof

of (a). According to the proposition ( Scaling equation ), We need to extend this relationship to any scale and location. We use the formula ( Property of scale and transport 7 ). Thus Therefore,

Proof

of (b). The wavelet is given by the equation thus We use the formula ( Property of scale and transport 7 ). Therefore, We calculate

Proof

of (c). According to the definition ( Approximation and detail operators ) and propositions ( Existence of orthonormal wavelet bases 1 ), ( Existence of orthonormal wavelet bases 2 ) We put into and substitute the terms We substitute and . Thus

 Notation. Index. Contents.