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.
 a. Quadrature mirror filter (QMF) conditions.
 b. Recovering scaling function from auxiliary function. Cascade algorithm.
 c. Recovering MRA from 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.

## Quadrature mirror filter (QMF) conditions.

efinition

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

(a) ,

(b) .

Proposition

(QMF property 1) Let and defined by the formula ( Definition of g_k ), is defined by and are defined in ( Approximation and detail operators 3 ). Then the following statements are equivalent:

(a) ,

(b) ,

(c) ,

(d) .

Proof

We calculate We use the proposition ( Parseval equality ). Now scale from to and use the proposition ( Fourier series on unit interval ).

Proof

and . The proof is repeated application of the proposition ( Interaction of approximation and detail operators with Fourier transform ).

Proposition

(QMF property 2) Let is QMF and defined by the formula ( Definition of g_k ) then

(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f) ,

(g) .

Proof

By the condition ( QMF conditions )-(a) and hence (a).

We have

By the condition ( QMF conditions ), We substitute into the last result and obtain (b).

The (c) is equivalent to (b), see the remark below.

The (d),(e),(f) and (g) are consequences of the proposition ( QMF property 1 ) and definitions ( Approximation and detail operators 3 ).

Remark

The statements of the proposition ( QMF property 2 ) have the following interpretation in terms of -geometry.

(a). .

(b). .

(c). .

(d). .

(e). if then .

(f).

(g). .

Proof

We will be using results of the section ( Fourier analysis in Hilbert space ) and the properties of scale and transport from the section ( Elementary definitions of wavelet analysis ) without further notice.

We prove (a):

We prove (b) with the same calculation as (a):

We prove (c): hence

To prove (d) we calculate The cases (e) and (f) are similar to (d). (g) is a collection of previous statements.

 Notation. Index. Contents.