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.
 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.

## Elementary definitions of wavelet analysis.

e already introduced the mesh in the previous section.

Given , the index is called the "location index" and the index is called the "scale" index.

Definition

(Scale and transport operators) We introduce the operations and according to the rules

Note that We introduce the notations , , , , :

Definition

(Scale and transport operators 2) We introduce the operations and according to the rules:

Note that Read the above " spt (see ) iff spt (see )". Consequently Similarly, We conclude

 (Property of scale and transport 1)
 (Property of scale and transport 2)
 (Property of scale and transport 3)

In the following sections we frequently use Fourier transform as well as the operations . We investigate the interaction:

 (Property of scale and transport 4)

We make a change ,

 (Property of scale and transport 5)

 (Property of scale and transport 6)

We investigate transposition of and :

 (Property of scale and transport 7)

Definition

(Orthonormal wavelet basis) The set of functions of the form , is called "orthonormal wavelet basis" if it is orthonormal:

and constitutes a basis in .

 Notation. Index. Contents.