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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.
a. Quadrature mirror filter (QMF) conditions.
b. Recovering scaling function from auxiliary function. Cascade algorithm.
c. Recovering MRA from 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.
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.

Quadrature mirror filter (QMF) conditions.


efinition

(QMF conditions) For MATH we define $m_{0}$ as in the proposition ( Scaling equation ). Then MATH "is QMF" provided that both of the following are true:

(a) MATH ,

(b) MATH .

Proposition

(QMF property 1) Let MATH and $g_{k}$ defined by the formula ( Definition of g_k ), $m_{0}$ is defined by MATH and $G,H$ are defined in ( Approximation and detail operators 3 ). Then the following statements are equivalent:

(a) MATH ,

(b) MATH ,

(c) MATH ,

(d) MATH .

Proof

MATH We calculate MATH We use the proposition ( Parseval equality ). MATH MATH MATH Now scale $z$ from MATH to $\left[ 0,1\right] $ and use the proposition ( Fourier series on unit interval ).

Proof

MATH and MATH . The proof is repeated application of the proposition ( Interaction of approximation and detail operators with Fourier transform ).

Proposition

(QMF property 2) Let MATH is QMF and $g_{k}$ defined by the formula ( Definition of g_k ) then

(a) MATH ,

(b) $\sum_{k}g_{k}=0$ ,

(c) MATH ,

(d) MATH ,

(e) MATH ,

(f) MATH ,

(g) MATH .

Proof

By the condition ( QMF conditions )-(a) MATH and MATH hence (a).

We have MATH

By the condition ( QMF conditions ), MATH We substitute $z=0$ into the last result and obtain (b).

The (c) is equivalent to (b), see the remark below.

The (d),(e),(f) and (g) are consequences of the proposition ( QMF property 1 ) and definitions ( Approximation and detail operators 3 ).

Remark

The statements of the proposition ( QMF property 2 ) have the following interpretation in terms of $L^{2}$ -geometry.

(a). MATH $\Leftrightarrow$ MATH .

(b). $\sum_{k}g_{k}=0$ $\Leftrightarrow$ MATH .

(c). $\sum_{k}g_{k}=0$ $\Rightarrow$ MATH .

(d). MATH $\Leftrightarrow$ MATH .

(e). if MATH then MATH $\Leftrightarrow$ MATH .

(f). MATH $\Leftrightarrow$ MATH

(g). MATH $\ \Leftrightarrow$ MATH $\ \Leftrightarrow$ MATH $\ \Leftrightarrow$ MATH .

Proof

We will be using results of the section ( Fourier analysis in Hilbert space ) and the properties of scale and transport from the section ( Elementary definitions of wavelet analysis ) without further notice.

We prove (a): MATH

We prove (b) with the same calculation as (a): MATH

We prove (c): MATH hence MATH

To prove (d) we calculate MATH MATH The cases (e) and (f) are similar to (d). (g) is a collection of previous statements.





Notation. Index. Contents.


















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