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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
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.

Existence of smooth compactly supported wavelets. Daubechies polynomials.


(Existence of smooth compactly supported wavelets) There exist smooth compactly supported wavelets calculated according to the following procedure:

1. Pick a number $n$ . It will determine the length of the scaling filter MATH and the size of support of $\phi$ , see the proposition ( Support of scaling function ).

2. Set MATH where the $C_{n}^{k}$ are the binomial coefficients, MATH .

3. Set MATH for some numbers MATH defined in the next step.

4. Match MATH to find MATH .

5. Set MATH

6. Use the proposition ( Cascade algorithm ) to find $\phi$ .

7. Use the proposition ( Scaling equation 2 ) to find $\psi$ .


The step 4 is possible for the following reasons. We have MATH for some numbers MATH and the sum is real-valued for $z\in\QTR{cal}{R}$ . Therefore the MATH has to be real-valued by orthogonality of MATH . We calculate MATH and the imaginary terms must vanish. Hence, MATH is a polynomial of MATH of degree $n-1$ and we can always do the match.


It follows from the step 5 and calculations of the previous remark that MATH is of the form MATH Thus MATH and the length of $spt$ $\phi$ is $2n-1$ , see the proposition ( Support of scaling function ).


Nothing prevents us from taking summation ranges MATH instead of MATH . The results would be the same up to a shift.


of existence. The key motivational statement is the proposition ( Recovering MRA from auxiliary function 1 ). According to the propositions ( Sufficient conditions for vanishing moments ), ( Vanishing moments vs decay at infinity ), ( QMF property 1 ) we are looking for a function MATH where the sequence MATH is finite due to compactness of support and real because we seek to build an MRA with real valued $\phi$ and thus have the proposition ( Scaling equation ). The MATH is sought to satisfy the following conditions: MATH MATH MATH MATH for some finite sequence of numbers MATH .

We have MATH

Let MATH We must have $c_{k}=c_{-k}$ so that MATH , MATH . Hence MATH for some finite set MATH . Also MATH MATH Hence, the requirement MATH takes the form MATH for some polynomial MATH . Therefore, we are looking for MATH that satisfies MATH We find such MATH by performing the following calculation: MATH We make change $m=2n-1-k$ in the second sum, MATH , MATH MATH


on smoothness. According to the proposition ( Reproduction of polynomials 4 ), there exist numbers MATH s.t. MATH Also, MATH

Thus, the number of discontinuities of $\phi$ and its derivatives has to be finite (by compactness of support of $\phi$ in $\left( \#\right) $ ).

Some insight into degree of smoothness is given by the proposition ( Smoothness of compactly supported wavelets with vanishing moments ). More precise investigation is given in the source [Daubechies1992] .The grows of regularity with $n$ is rather slow. $\phi,\psi$ are continuously differentiable for $n=3$ . For large $n$ regularity grows like $0.2n$ .


The following Mathematica script implements the procedure of the proposition ( Existence of smooth compactly supported wavelets ). The results agree with Python's pywt module.


CC[k_, n_] := Binomial[n, k]

Pnm1[n_, y_] :=

Expand[Sum[CC[k, 2*n - 1]*y^k*(1 - y)^(n - 1 - k), {k, 0, n - 1}]]




x1 = Cond[n, z]

x2 = TrigToExp[ComplexExpand[x1]]

d = Exponent[x2, Exp[I*Pi*z]]

x3 = Collect[x2*Exp[d*I*Pi*z], Exp[I*Pi*z]]

L=CoefficientList[x3, Exp[I*Pi*z]]

eqs=Map[Function[x, x == 0], L]

eqs2 = Map[Function[x, Im[a[x]] == 0], Range[0, n-1]]

eqs3={ A[n,0]==1 }

vars = Map[Function[x, a[x]], Range[0, n-1]]

sol=NSolve[Join[eqs, eqs2,eqs3], vars]

x4 = m0[n, z] /. sol[[1]]

x5=Collect[x4, Exp[I*Pi*z]]

LL=CoefficientList[x5, Exp[-2*I*Pi*z]]


Notation. Index. Contents.

Copyright 2007