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.

## Vanishing moments vs approximation.

roposition

(Vanishing moments vs approximation) Assume that

1. a function ,

2. the derivative is bounded on : for some ,

3. a function has compact support,

4. , ,

5.

then there exists a constant such that

Proof

We assume without loss of generality that see the notation of the section ( Elementary definitions of wavelet analysis ). We estimate directly: We use the formula ( Property of scale and transport 1 ) and the proposition ( Taylor decomposition in Schlomilch, Lagrange and Cauchy forms ). Note that is the middle point of . where . We use the condition (4). We use the condition (2). We use the formula ( Holder inequality ).

 Notation. Index. Contents.