Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.

## Multiresolution analysis.

e intend to provide a procedure for implementation of the following structure.

Definition

(Multiresolution analysis) We call "multiresolution analysis" (MRA) the sequence of subspaces , with the following properties.

1.

2. , .

3.

4.

5. such that and

The closures are taken in -norm.

See the definition ( Scale and transport operators 2 ) for and . The function is called the "scaling function".

Definition

(Approximation and detail operators) We introduce the "approximation operators" and "detail operators" according to the formulas

Proposition

(Subspace bases) Let be an MRA. The set is an orthonormal basis of for every .

Proof

Let . Then, according to the definition ( Multiresolution analysis )-4, . Hence, by the definition ( Multiresolution analysis )-4,5, is approximated in with any precision by linear combinations of . But preserves the -norm, see the formula ( Property of scale and transport 2 ). Therefore, is approximated by linear combinations of .

The orthonormality follows from the definition ( Multiresolution analysis )-5 and the formulas ( Property of scale and transport 2 ) and ( Property of scale and transport 3 ).

 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.
 Notation. Index. Contents.