Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 decay at infinity.

roposition

(Vanishing moments vs decay at infinity) Suppose that and the collection is -orthogonal with respect to the scale index:

For any if and then , .

Proof

We provide a proof by induction in .

In the case we have for : We use the proposition ( Basic properties of Fourier transform )-3. We use the formula ( Property of scale and transport 6 ). Hence, We set and choose so that for some to be determined later. By the inclusion and the proposition ( Basic properties of Fourier transform )-5 By the inclusion and the proposition ( Dominated convergence theorem ), We choose so that then

For general , we assume that the statement is already proven for . We have By inclusion we have thus We continue The integral in the above estimate for is bounded as by inclusion . By induction assumption , . Hence, we cancel and pass to obtain

 Notation. Index. Contents.