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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.


efinition

(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

Definition

(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.

Proposition

(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 .

Proof

Direct verification.

Proposition

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

Proof

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

Proposition

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

Proof

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

Proposition

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

Proof

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

Proposition

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

Proof

Direct consequence of the propositions ( Approximation and detail operators 4 ), ( Fourier transform of convolution ),( Interaction of downsampling with Fourier transform ),( Interaction of upsampling with Fourier transform ) and ( Interaction of tilde with Fourier transform ).





Notation. Index. Contents.


















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