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.
 a. Auxiliary function of OST.
 b. Scaling equation for wavelet.
 c. Existence of 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.

## Auxiliary function of OST.

roposition

(Scaling equation 3) Let be an MRA and is the auxiliary function (as in the definition ( Scaling filter and auxiliary function )). Then the function satisfies the relationship

Proof

According to the proposition ( OST property 1 ), we have for We use the proposition ( Scaling equation ). We separate even and odd -terms. According to the definition of , see the proposition ( Scaling equation ), . We use the proposition ( OST property 1 ) again.

 Notation. Index. Contents.