Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 MATH according to the relationships MATH where the operation $\ast$ denotes convolution. We define MATH by MATH

Proposition

(Properties of spline functions)

1. MATH , MATH .

2. MATH .

3. MATH is $n$ -th degree polynomial for any $k,n$ .

4. MATH , MATH

Proof

1,2 and 3 are direct verification by induction.

We have MATH MATH We repeat calculations of the proposition ( Scaling equation ) and arrive to MATH According to the proposition ( Basic properties of Fourier transform )-6, MATH 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 MATH is verified via the proposition ( Shifted Fourier transform equality ): MATH Due to the proposition ( Properties of spline functions )-1, only the terms MATH 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.


















Copyright 2007