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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 scaling function from auxiliary function. Cascade algorithm.


iven MATH satisfying the definition ( QMF conditions ) on may construct the auxiliary function (see the proposition ( Scaling equation )) MATH If the scaling function $\phi$ exists then it would satisfy MATH Then, assuming convergence of the product, MATH

Proposition

(Convergence of product) If the series MATH converge absolutely then the product $\prod _{n}z_{n}$ converges absolutely.

Proof

We calculate MATH Since MATH starting from some $n$ MATH and MATH for some $c=const$ .

Proposition

(Finite QMF convergence) Let MATH be a finite QMF. Then for any $R>0$ the product MATH converges absolutely and in MATH .

Proof

We estimate MATH and utilize the proposition ( QMF property 2 )-a: MATH MATH MATH Therefore MATH Since MATH is finite, the sum MATH has some finite value MATH . We use the proposition ( Convergence of product ), convergence of MATH is investigated by looking at MATH and the statement follows.

Proposition

(Cascade algorithm) Let MATH be a finite QMF. Define operator $A$ : MATH and let MATH be a sequence such that MATH Then MATH

Proof

Note that by proposition ( QMF property 2 )-d and formulas ( Property of scale and transport 2 ),( Property of scale and transport 3 ), MATH Thus MATH

By calculations of the proof of the proposition ( Scaling equation ), MATH thus MATH and by the proposition ( Finite QMF convergence ) MATH Combine this with $\left( \#\right) $ , proposition ( Basic properties of Fourier transform ) and the desired conclusion follows.

Proposition

A QMF MATH is finite if and only if $\phi$ has finite support.

Proof

( $\Rightarrow$ ). $\phi$ is obtained by iterations MATH of the proposition ( Cascade algorithm ). The scaling within the operator $A$ halves the support of $\eta_{m}$ and the summation increases it linearly. Such operation, when applied repeatedly, leads to compact support.

( $\Leftarrow$ ) If $\phi$ has compact support then MATH is finite by the defining formula of the proposition ( Scaling equation ).





Notation. Index. Contents.


















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