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.
 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.

## Discrete wavelet transform.

n this section we frequently assume the following setup.

Condition

(Discrete wavelet transform setup) There is an MRA (see the definition ( Multiresolution analysis )), scaling function , sequence (see the proposition ( Scaling equation )), wavelet (see the proposition ( Existence of orthonormal wavelet bases 2 )) and operators , (see the definition ( Approximation and detail operators )).

We introduce the notation

 (Definition of g k)

 a. Recursive relationships for wavelet transform.
 b. Properties of sequences h and g.
 c. Approximation and detail operators.
 Notation. Index. Contents.