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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.
a. Vanishing moments vs decay at infinity.
b. Vanishing moments vs approximation.
c. Sufficient conditions for vanishing moments.
d. Reproduction of polynomials.
e. Smoothness of compactly supported wavelets with vanishing moments.
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.

Reproduction of polynomials.


roposition

(Reproduction of polynomials 1) Let MATH be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. $\phi$ is compactly supported and (therefore) MATH is finite.

2. MATH , $k=0,1,...,n-1$ .

Then MATH

Proof

According to the proposition ( Scaling equation ), MATH thus MATH Let $j$ be odd, $j=2s+1$ then MATH We apply the proposition ( Sufficient conditions for vanishing moments )-b MATH

For $j$ even we have $j=2^{p}s$ , for some integer $p$ and an odd integer $s$ . We use the proposition ( Scaling equation ) $p$ times: MATH thus MATH and we repeat the calculation of the odd case: MATH

Remark

Note that MATH is a 1-periodic function. Hence, we may consider the Fourier coefficients of MATH : MATH We make a change $x+k=y$ . MATH Under conditions of the last proposition, MATH . Thus MATH

Proposition

(Reproduction of polynomials 2)Let MATH be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. $\phi$ is compactly supported and (therefore) MATH is finite,

2. MATH , $k=0,1,...,n-1$ .

Then MATH for $k=0,1,...,n-1$ .

Proof

The function MATH has period 1. We evaluate an $m$ -th Fourier coefficient: MATH MATH We apply the proposition ( Reproduction of polynomials 1 ). MATH Thus, by section ( Fourier series section ), MATH

Proposition

(Reproduction of polynomials 3) Let MATH be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. $\phi$ is compactly supported and (therefore) MATH is finite,

2. MATH , $k=0,1,...,n-1$ .

Then there exist polynomials MATH of degree $k-1$ such that MATH for $k=0,1,...,n-1$ .

Proof

We introduce the convenience notation MATH then the proposition ( Reproduction of polynomials 2 ) provides MATH For $k=0$ we have MATH For $k=1$ we have MATH so that MATH For $k=2$ we have MATH MATH We use results of previous steps. MATH We continue similarly for all $k$ up to $n-1$ .

Proposition

(Reproduction of polynomials 4) Let MATH be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. $\phi$ is compactly supported and (therefore) MATH is finite,

2. MATH , $k=0,1,...,n-1$ .

Then there are numbers MATH such that MATH for $k=0,1,...,n-1$ .

Proof

Take linear combination of equations from the proposition ( Reproduction of polynomials 3 ).





Notation. Index. Contents.


















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