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Quantitative Analysis
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Numerical Analysis
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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.

Dual wavelets.


efinition

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

Proposition

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

Proof

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

Proposition

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

(a) MATH .

(b) MATH is biorthogonal to MATH .

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

(d) MATH , MATH .

(e) MATH we have MATH and MATH .

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

We used the notation MATH

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 MATH using the proposition ( Scaling equation 5 ): MATH We aim to use 1-periodicity of $m_{1}\,$ , hence we separate even and odd terms: MATH The $\phi$ comes from a GMRA, hence MATH is a Riesz basis and ( Frame property 2 )-2 applies to each MATH : MATH Hence, we need an estimate for $m_{0}$ . The $\phi$ already has the frame property by ( Frame property 2 )-2: MATH We repeat the same calculation for $\phi$ starting from the left inequality: MATH thus MATH Similarly, starting from MATH we get MATH Combining these two inequalities with $\left( \#\right) $ completes the proof of (c).

Proof

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

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

We now verify orthogonality across scales. Let $d_{1}<d_{2}$ . We have MATH MATH By the formula ( Property of scale and transport 2 ) and (d) we have MATH Hence MATH for any $d_{1}<d_{2}$ and any $k_{1},k_{2}$ .





Notation. Index. Contents.


















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