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.

## Orthogonality across scales.

n this section we present a procedure for construction of dual wavelets with orthogonality across scales. The disadvantage of this construction is non-compactness of support for one of the wavelets . In the section ( Compactly supported smooth biorthogonal wavelets section ) we present a pair with compact support but without orthogonality across scales (property (d) below).

Proposition

(Existence of wavelets with orthogonality across scales) Let be a GMRA with a compactly supported scaling function . There exists a scaling function : and the associated dual GMRA is equipped with the following properties.

(a) .

(b) , when .

We use notation , the closure is in .

(c) , , .

(d)

(e) such that

Proof

of existence of GMRA and (a).

According to the proposition ( Property of transport 2 ), the function has the form for some numbers and finite . By the proposition ( Frame property 1 ) and definition ( Generalized multiresolution analysis )-5, is separated from 0. Therefore the function is in . It also has period 1. Therefore, we write for some .

We introduce according to the relationship By the proposition ( Existence of biorthogonal basis 1 ) we have biorthogonality of and . In addition hence is a Riesz basis on by the proposition ( Frame property 2 ).

We take inverse Fourier transform of : thus Therefore The inverse inclusion is a consequence of biorthogonality of and . Indeed, suppose there is an such that . We have for some numbers . Because of biorthogonality, Let Since we must have Let are numbers such that We have We obtained a contradiction. Hence, and by the formula ( Property of scale and transport 2 ) and definition ( Generalized multiresolution analysis )-4 we have

Proof

of (b),(c),(d). We define and the same way we did in the definition ( Dual wavelets ). We get biorthogonality of and by the proposition ( Dual wavelets properties )-f. We also have ( Dual wavelets properties )-a,d,e: and (a) of the present proposition: From and we get and Then, by the formula ( Property of scale and transport 2 ), and by Then, by the formula ( Property of scale and transport 2 ), Thus (b) and (c). The (d) follows because, by proposition ( Dual wavelets properties )-f, and are bases in .

Proof

of (e). For every we have where For each we have by the proposition ( Dual wavelets properties )-c and formula ( Property of scale and transport 2 ) we sum the above in and combine with and to obtain (e).

 Notation. Index. Contents.