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 we define "shift operator" acting on sequences:

2. We define "downsampling operator" as follows:

3. We define "upsampling operator":

4. We define "convolution":

5. We define the "tilde operation":

Definition

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

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) , ,

(b) The adjoint operators act as follows:

(c) , ,

(d) ,

(e) ,

(f) ,

(g) .

Proof

Direct verification.

Proposition

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

Proof

We use the proposition ( Fourier series on unit interval ): Make change . Make change in the second integral. Therefore, by the proposition ( Fourier series on unit interval ), we must have

Proposition

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

Proof

We verify directly: Make change .

Proposition

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

Proof

We calculate We make change .

Proposition

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

 Notation. Index. Contents.