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.

## Approximation by system of translates.

roposition

(OST property 2) Let be OST. Then iff The closure is taken in .

Proof

of . Suppose takes place. According to the proposition ( Main property of Fourier decomposition ), We take Fourier transform of both sides: and use the formula ( Property of scale and transport 4 ): Thus we have with . We have by the proposition ( Parseval equality ).

Proof

of . Suppose takes place. Then we perform the inverse Fourier transform and the above calculations performed in reverse order lead to .

 Notation. Index. Contents.