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.

## Properties of sequences h and g.

roposition

(Integral of scaling function) Let is a scaling function of an MRA. Then

Proof

By the definition ( Multiresolution analysis )-1,2 we have We calculate the last term using the proposition ( Basic properties of Fourier transform )-3: We use the formula ( Property of scale and transport 6 ): Using the propositions ( Dominated convergence theorem ) and ( Fubini theorem ), The collection is an orthonormal basis in . We use the proposition ( Parseval equality ). We use the proposition ( Basic properties of Fourier transform )-4,5. Thus

Proposition

(Integral of wavelet) Let is a scaling function of an MRA and is the associated wavelet, see the proposition ( Scaling equation 2 ). We have

Proof

According to the proposition ( Scaling equation 2 ), Thus According to the proposition ( Scaling equation 3 ), Thus where According to the proposition ( Scaling equation ), an according to the proposition ( Integral of scaling function ) thus

Proposition

(Properties of sequences h and g) We assume the setup ( Discrete wavelet transform ), and is real-valued. Then

(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f) .

Proof

of (a). Within the proof the proposition ( Integral of wavelet ) we saw

Proof

of (b). According the proposition ( Integral of wavelet ), Change of variable, and then . We use the proposition ( Integral of scaling function ).

Proof

of (c),(d),(e). We use orthonormality of for every : We use the formula ( Property of scale and transport 7 ). The property (d) follows similarly from orthonormality of for every , see the proposition ( Existence of orthonormal wavelet bases 2 ). The property (e) follows by considering , see the proposition ( Existence of orthonormal wavelet bases 1 )-c.

Proof

of (f). We combine the proposition ( Recursive relationships for wavelet transform )-c with ( Recursive relationships for wavelet transform )-a,b: We now express and in terms of using ( Recursive relationships for wavelet transform )-a,b. Thus

 Notation. Index. Contents.