Quantitative 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 auxilliary function. Cascade algorithm.
c. Recovering MRA from auxilliary 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.

Recovering MRA from auxilliary function.


roposition

(Recovering MRA from auxilliary function 1) Suppose a function MATH satisfies the following conditions:

1. $m_{0}$ is given by a finite filter MATH : MATH

2. $m_{0}$ satisfies the conditions ( QMF conditions ).

Then there exists an MRA MATH such that the $\phi$ recovered from $m_{0}$ via MATH is a scaling function of MATH .

Proof

We form MATH Then the condition ( Multiresolution analysis )-4 and MATH are evident. The orthogonality part of ( Multiresolution analysis )-5 follows from the QMF conditions, see the proof of the proposition ( Scaling equation 3 ).

We now prove the condition ( Multiresolution analysis )-1.

By condition 1 and according to the section ( Fourier transform of delta function ), MATH For a general function $f$ , MATH thus MATH From the formula $\left( \#\right) $ we conclude MATH We take inverse Fourier transform of the above. The product becomes convolution: MATH Thus we have the condition ( Multiresolution analysis )-1: MATH

We now verify the condition ( Multiresolution analysis )-3. It suffices to show that MATH we have MATH We start with the case of function $f$ with compact support: MATH We estimate MATH MATH MATH for some MATH . Note that MATH when $k\rightarrow\infty$ . Then MATH and MATH converges because MATH . Thus MATH

The set of functions $f$ with compact support constitutes a dense set in $L^{2}$ and for each function on such dense set we have MATH Hence, the above convergence result extends to all MATH by the following standard argument. Let $g\in L^{2}$ and MATH have compact support and MATH , as $k\rightarrow\infty$ . MATH If MATH does not converge to zero then there is a subsequence MATH such that MATH is separated from zero. But then we arrive to contradiction because everything on the RHS of MATH can be arbitrarily small.

The final task is to verify the condition ( Multiresolution analysis )-2. We need to show MATH Note that MATH thus MATH or MATH According to the proposition ( Fourier transform of projection on span of translates ), MATH We use the formula ( Property of scale and transport 5 ). MATH We assume that the function $\hat{f}$ has compact support and prove the statement. Then the general case follows by density argument, similarly to the above. MATH MATH Note that if $d$ is large enough and $\hat{f}$ has compact support then the only non-zero terms are those with $r=k$ . MATH We make the change $y=z-k$ . MATH Note that the sum has structure of integral as MATH . The term $2^{d}y$ becomes zero, $2^{-d}k$ becomes the variable of integration: MATH where the last equality follows from the condition 2 and the propositions ( QMF property 1 )-b and ( Basic properties of Fourier transform )-3. We continue MATH We make a change $2^{-d}y=z$ . MATH Since MATH is finite, $\phi$ has compact support and, thus, $\hat{\phi}$ is infinitely smooth. Hence, MATH where MATH and thus MATH . Then, by $\left( \#\right) $ , condition 2 and the proposition ( QMF property 2 )-a, we have MATH . MATH We make the reverse change of variables. MATH We obtained MATH





Notation. Index. Contents.


















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