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.
 a. Scaling equation.
 b. Support of scaling function.
 c. Piecewise linear MRA, part 1.
 d. Orthonormal system of translates.
 e. Approximation by system of translates.
 f. Orthogonalization of system of translates.
 g. Piecewise linear MRA, part 2.
 h. Construction of MRA summary.
 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.

Orthogonalization of system of translates.

roposition

(OST property 3) Assume that

1. ,

2. has compact support,

3. there exist constants such that

 (Riesz basis condition)
then there exists a function such that

a.

b. is OST,

c. . The closure is taken in .

Proof

We look for in . Then (a) is trivial and (c) would be satisfied (see below). To get (b) we use the proposition ( OST property 1 ).

We set We need to achieve (a) because (see the formula ( Property of scale and transport 3 )) and we need to achieve (b) (see the proposition ( OST property 1 )). We take the Fourier transform of : We use the formula ( Property of scale and transport 4 ): Hence Therefore, we would like to choose to satisfy according to the rule Hence, we have if we establish that the RHS is in (see the sections ( Fourier series ) and ( Fourier analysis in Hilbert space )). But this is evident from the conditions (1),(2) and (3).

It remains to prove that with implies (c). The statement is trivial. We need to prove . Let so that We seek such that or

Thus we need to have and under assumptions that . Furthermore, by the condition (2), the summation in and the number of is finite. Hence, is finite dimensional linear equation. The matrix of has the form for some index ranges and of the same length depending on diameter of . Since has only a finite number non-zero coordinates, the matrix is semi-diagonal. Hence, and always have a solution.

 Notation. Index. Contents.