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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 $\psi,\tilde{\psi}$ . In the section ( Compactly supported smooth biorthogonal wavelets section ) we present a pair $\psi,\tilde{\psi}$ with compact support but without orthogonality across scales (property (d) below).

Proposition

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

(a) MATH .

(b) MATH , MATH when $~d_{1}\not =d_{2}$ .

We use notation MATH , the closure is in MATH .

(c) $W_{d}\bot V_{d}$ , MATH , MATH .

(d) MATH

(e) MATH such that MATH

Proof

of existence of GMRA and (a).

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

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

We take inverse Fourier transform of $\left( \#\right) $ : MATH thus MATH Therefore MATH The inverse inclusion MATH is a consequence of biorthogonality of MATH and MATH . Indeed, suppose there is an $f\in V_{0}$ such that MATH . We have MATH for some numbers MATH . Because of biorthogonality, MATH Let MATH Since MATH we must have MATH Let MATH are numbers such that MATH We have MATH We obtained a contradiction. Hence, MATH and by the formula ( Property of scale and transport 2 ) and definition ( Generalized multiresolution analysis )-4 we have MATH

Proof

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

Proof

of (e). For every $f$ we have MATH where MATH For each $d\in\QTR{cal}{Z}$ we have by the proposition ( Dual wavelets properties )-c and formula ( Property of scale and transport 2 ) MATH we sum the above in $d$ and combine with MATH and MATH to obtain (e).





Notation. Index. Contents.


















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