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.

## Spline functions.

efinition

(Spline functions) We define the family according to the relationships where the operation denotes convolution. We define by

Proposition

(Properties of spline functions)

1. , .

2. .

3. is -th degree polynomial for any .

4. ,

Proof

1,2 and 3 are direct verification by induction.

We have We repeat calculations of the proposition ( Scaling equation ) and arrive to According to the proposition ( Basic properties of Fourier transform )-6, and we arrive to 4.

Remark

The following Mathematica script evaluates spline functions.

n=3

Clear[x]

Clear[y]

B[0] = Piecewise[{{1, -1/2 < x && x < 1/2}, {0, x <= -1/2}, {0, 1/2 <= x}}]

For[i=1,i<=n,i=i+1, B[i]=Convolve[B[i-1] /. x -> y, B[0] /. x -> y, y, x]]

B[n_, z_] := B[n] /. x -> z

Btilde[n_,x_]:=B[n,x-(n+1)/2]

Remark

The applicability of the proposition ( Existence of biorthogonal basis 1 ) to the functions is verified via the proposition ( Shifted Fourier transform equality ): Due to the proposition ( Properties of spline functions )-1, only the terms participate in the RHS sum. Therefore, the following continuation of the above Mathematica script checks the applicability of the proposition ( Existence of biorthogonal basis 1 ).

Clear[x]

Ck = Map[Function[k,

NIntegrate[

Btilde[n, x]*Btilde[n, x - k], {x, -\[Infinity], +\[Infinity]}]],

Range[-n, n]]

Sgma[x_] := Sum[Ck[[k + n + 1]]*Exp[-2*Pi*I*k*x], {k, -n, n}]

Plot[Re[Sgma[x]], {x, 0, 1}]

 Notation. Index. Contents.