Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
a. Recursive relationships for wavelet transform.
b. Properties of sequences h and g.
c. Approximation and detail operators.
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.

Approximation and detail operators.


(Shift and sampling operators)

1. For any $m\in\QTR{cal}{Z}$ we define "shift operator" $\tau_{m}$ acting on sequences: MATH

2. We define "downsampling operator" $\downarrow$ as follows: MATH

3. We define "upsampling operator": MATH

4. We define "convolution": MATH

5. We define the "tilde operation": MATH


(Approximation and detail operators 3) Given MATH we define MATH as in the formula ( Definition of g_k ) and introduce the operators $H$ and $G$ : MATH

Compare with the proposition ( Recursive relationships for wavelet transform )-a,b for motivation.


(Approximation and detail operators 4)

In context of the definition ( Approximation and detail operators 3 ) we have

(a) MATH , MATH ,

(b) The adjoint operators act as follows: MATH

(c) MATH , MATH ,

(d) MATH $\Leftrightarrow$ $H^{\ast}H=I$ ,

(e) MATH $\Leftrightarrow$ $G^{\ast}G=I$ ,

(f) MATH $\Leftrightarrow$ MATH ,

(g) MATH $\Leftrightarrow$ MATH .


Direct verification.


(Interaction of downsampling with Fourier transform) Given MATH we introduce MATH Then MATH


We use the proposition ( Fourier series on unit interval ): MATH Make change $2z=y$ . MATH Make change $y=x+1$ in the second integral. MATH Therefore, by the proposition ( Fourier series on unit interval ), we must have MATH


(Interaction of upsampling with Fourier transform) For a MATH we have MATH


We verify directly: MATH Make change $k=2p$ . MATH


(Interaction of tilde with Fourier transform) For a MATH we have MATH


We calculate MATH We make change $p=-k$ . MATH


(Interaction of approximation and detail operators with Fourier transform) For a MATH we have MATH

Notation. Index. Contents.

Copyright 2007