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.

## Smoothness of compactly supported wavelets with vanishing moments.

roposition

(Smoothness of compactly supported wavelets with 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.

2. , , for some .

Then where and comes from the proposition ( Sufficient conditions for vanishing moments )-c.

Remark

The above proposition is to be used with the proposition ( Basic properties of Fourier transform )-7.

Proof

the proof may found in [Mallat] , proposition 7.6, p. 287.

 Notation. Index. Contents.