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.
 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.
 a. Biorthogonal bases.
 b. Riesz bases.
 c. Generalized multiresolution analysis.
 d. Dual generalized multiresolution analysis.
 e. Dual wavelets.
 f. Orthogonality across scales.
 g. Biorthogonal QMF conditions.
 h. Vanishing moments for biorthogonal wavelets.
 i. Compactly supported smooth biorthogonal wavelets.
 j. Spline functions.
 k. Calculation of spline biorthogonal wavelets.
 l. Symmetric biorthogonal wavelets.
 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.

## Biorthogonal QMF conditions.

otivation and notation of this section is based on the section ( QMF section ).

Definition

(Biorthogonal QMF conditions) For we define as in the proposition ( Scaling equation ). Then "are QMF pair" provided that both of the following are true:

(a) ,

(b) .

Proposition

(Biorthogonal QMF property 1) Let , are from the definition ( Dual wavelets ), is defined by and are defined in ( Approximation and detail operators 3 ) and are similarly defined by replacement , . Then the following statements are equivalent:

(a) ,

(b) (or, equivalently, ),

(c) ,

(d) .

Proof

The proof is very similar to the proof of the proposition ( QMF property 1 ). For example, let us show equivalence of (a) and (b). We calculate We use the proposition ( Parseval equality ).

Proposition

(Biorthogonal QMF property 2) Let are QMF pair, are from the definition ( Dual wavelets ) then

(a) ,

(b) ,

(d) ,

(e) ,

(f) ,

(g) .

Proof

The proof is similar to the proof of the proposition ( QMF property 2 ).

 Notation. Index. Contents.