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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.
a. Scaling equation.
b. Support of scaling function.
c. Piecewise linear MRA, part 1.
d. Orthonormal system of translates.
e. Approximation by system of translates.
f. Orthogonalization of system of translates.
g. Piecewise linear MRA, part 2.
h. Construction of MRA summary.
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.
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.

Orthonormal system of translates.


efinition

(Orthonormal system of translates) For a function MATH we form MATH . If MATH has the property MATH then we call it "orthonormal system of translates" (OST).

Note that MATH and the calculation goes in either direction. Thus the OST may be redefined with the requirement

MATH (OST property 0)

Proposition

(Property of transport 1) For any function MATH we have MATH where $\hat{g}$ is the Fourier transform of $g$ and the integral is a $\left( -k\right) $ -th Fourier coefficient of a 1-periodic function MATH .

Proof

Fourier transform preserves $L^{2}$ geometry, see ( Parseval equality ) and ( Basic properties of Fourier transform )-3. Hence, MATH We use the formula ( Property of scale and transport 4 ). MATH Make the change $y=z-n$ . MATH We use the proposition ( Fubini theorem ), MATH . MATH

Proposition

(Property of transport 2) If MATH has compact support then the function MATH has the form MATH for some numbers MATH and finite $k_{1},k_{2}$ .

Proof

According to the proposition ( Property of transport 1 ), MATH Since $g$ has compact support, MATH for a finite number of $k$ .

Proposition

(OST property 1) The set MATH is OST iff MATH

Proof

According to the proposition ( Property of transport 1 ), MATH MATH The Fourier transformation is 1-1 in MATH and the collection MATH is orthonormal in MATH . Hence, MATH if and only if MATH and since the summation in $n$ is $-\infty$ to $+\infty$ , the above is true for MATH .





Notation. Index. Contents.


















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