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.
 a. Vanishing moments vs decay at infinity.
 b. Vanishing moments vs approximation.
 c. Sufficient conditions for vanishing moments.
 d. Reproduction of polynomials.
 e. Smoothness of compactly supported wavelets with vanishing moments.
 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.

## Sufficient conditions for vanishing moments.

roposition

(Sufficient conditions for vanishing moments) Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite, where the sequence is defined in the proposition ( Scaling equation ).

Then the following statements are equivalent:

(a) , , for some ,

(b) , ,

(c) there is a representation for some finite sequence ,

(d) , .

Proof

(a) (b). According to the proposition ( Scaling equation 2 ) and according to the proposition ( Integral of scaling function ), Therefore can be zero if and only if .

We continue We use the previous result : and conclude .

We continue similarly for any : for any derivative there will be a term and the rest of the terms are zero at by results of the previous steps.

Proof

(b) (c) Set By condition (1) the is defined for all except, possibly, . We have Hence, if (b) takes place then We do Taylor decomposition, (see the proposition ( Taylor decomposition in Schlomilch, Lagrange and Cauchy forms )): Thus, by (b) and , The then must be of the form for a finite sequence . This is (c).

The calculation may be performed in opposite direction: assume (c), then the has the form and we do argument in reverse arriving to (b).

Proof

(b) (d) The proof is a direct verification. We use finiteness of to differentiate termwise and substitute into (b).

 Notation. Index. Contents.