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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.

Recovering MRA from auxiliary function.


roposition

(Recovering MRA from auxiliary 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.

From the formula $\left( \#\right) $ we conclude MATH We would like to take inverse Fourier transform of $\left( @\right) $ .

By condition 1 and according to the section ( Fourier transform of delta function ), MATH For a general function $f$ , MATH thus MATH We now take the inverse Fourier transform of the equality $\left( @\right) $ . 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 First, we consider function $f$ with compact support: MATH We estimate MATH MATH MATH 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 verification of condition ( Multiresolution analysis )-2 may be found in [Mallat] , page 276.





Notation. Index. Contents.


















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