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.

## Dual wavelets.

efinition

(Dual wavelets) Let and be dual GMRA (see the definition ( Dual GMRA )) and are the corresponding sequences defined in the proposition ( Scaling equation ). We define the "wavelet" and "dual wavelet" according to the formulas

Proposition

(Scaling equation 5) In context of the definition ( Dual wavelets ) we have where

Proof

See the proof of the proposition ( Scaling equation 2 ).

Proposition

(Dual wavelets properties) and be dual GMRAs and the corresponding wavelets. Then the following holds.

(a) .

(b) is biorthogonal to .

(c) is a Riesz basis for , is a Riesz basis for .

(d) , .

(e) we have and .

(f) The collections and are biorthogonal and are bases in .

We used the notation

The properties (a),(b),(d),(e) are proven almost exactly as the similar statements of the propositions ( Existence of orthonormal wavelet bases 1 ) and ( Existence of orthonormal wavelet bases 2 ). The (c) and (f) are proven below.

Proof

(c) The method of the proof if to verify condition of the proposition ( Frame property 2 )-1. We calculate using the proposition ( Scaling equation 5 ): We aim to use 1-periodicity of , hence we separate even and odd terms: The comes from a GMRA, hence is a Riesz basis and ( Frame property 2 )-2 applies to each : Hence, we need an estimate for . The already has the frame property by ( Frame property 2 )-2: We repeat the same calculation for starting from the left inequality: thus Similarly, starting from we get Combining these two inequalities with completes the proof of (c).

Proof

(f) The inclusion follows from (e) and the definition ( Generalized multiresolution analysis ).

It remains to verify that and are biorthogonal. First, we verify for the same scale index : We use the formula ( Property of scale and transport 2 ). We use (b).

We now verify orthogonality across scales. Let . We have By the formula ( Property of scale and transport 2 ) and (d) we have Hence for any and any .

 Notation. Index. Contents.