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.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 a. Biorthogonal bases.
 b. Riesz bases.
 c. Generalized multiresolution analysis.
 d. Dual generalized multiresolution analysis.
 e. Dual wavelets.
 f. Orthogonality across scales.
 g. Biorthogonal QMF conditions.
 h. Vanishing moments for biorthogonal wavelets.
 i. Compactly supported smooth biorthogonal wavelets.
 j. Spline functions.
 k. Calculation of spline biorthogonal wavelets.
 l. Symmetric biorthogonal wavelets.
 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 for biorthogonal wavelets.

roposition

(Integral of scaling function 2) Let is a scaling function of a GMRA. Then

Proof

is similar to the proof of the proposition ( Integral of scaling function ).

The proposition ( Sufficient conditions for vanishing moments ) extends to a pair of dual GMRAs as follows.

Proposition

(Sufficient conditions for vanishing moments 2) Let and be an pair of dual GMRAs, scaling functions and wavelets (see the definition ( Dual wavelets )). Assume that

1. are compactly supported and (therefore) are finite.

Then the following statements are equivalent:

(a) , , for some .

(b) , ,

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

Proof

(a) (b). According to the proposition ( Scaling equation 5 ) and according to the proposition ( Integral of scaling function 2 ), Therefore can be zero if and only if .

We continue as in the proof of the proposition ( Sufficient conditions for vanishing moments ).

Proof

(b) (c) Set By condition (1) the is defined for all except, possibly, . We have and continue as in the proof of the proposition ( Sufficient conditions for vanishing moments ).

 Notation. Index. Contents.