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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.
a. Auxiliary function of OST.
b. Scaling equation for wavelet.
c. Existence of 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 orthonormal wavelet bases.


roposition

(Existence of orthonormal wavelet bases 1) Let MATH be an MRA and a function MATH satisfies the following conditions

a. $\psi\in V_{1}$ .

b. MATH is OST.

c. $W_{0}\perp V_{0}$ in $L^{2}$ .

d. MATH , $Q_{0}f\in W_{0}$ .

Then MATH is an orthogonal basis in MATH .

See the definition ( Approximation and detail operators ) for the notation $Q_{0}$ .

Proof

Orthogonality with respect to the index $k$ follows trivially from (b). To show orthogonality with respect to the index $d$ we apply the operation $S_{d}^{\ast}$ to (a),(c) and use the definition ( Multiresolution analysis )-4 to conclude MATH Thus for any pair $\psi_{d_{1},k_{1}}$ , $\psi_{d_{2},k_{2}}$ , $d_{1}<d_{2}$ we have MATH hence MATH

We use (d) and the definition ( Multiresolution analysis )-2 to prove that MATH is a basis. Indeed, we have MATH and MATH

Proposition

(Existence of orthonormal wavelet bases 2) Let MATH be an MRA (see the definition ( Multiresolution analysis )) with the scaling function $\phi$ and MATH is the sequence defined in the proposition ( Scaling equation ). Set MATH Then MATH is an orthonormal wavelet basis.

Proof

We prove the statement by verifying the conditions of the proposition ( Existence of orthonormal wavelet bases 1 ). The condition (a) is trivial.

We verify the condition (b) directly: MATH Note that the coefficients MATH have to obey orthogonality of MATH : MATH Therefore MATH

We verify the requirement ( Existence of orthonormal wavelet bases 1 )-(c) next. According to the proposition ( Basic properties of Fourier transform ), MATH We use the formula ( Property of scale and transport 4 ). MATH We use the propositions ( Scaling equation ) and ( Scaling equation 2 ). MATH MATH We make a change $z=y+m$ . MATH We separate even and odd terms. MATH MATH We use the properties MATH , MATH . MATH MATH MATH MATH We use the proposition ( OST property 1 ). MATH We use the proposition ( Scaling equation 2 ), MATH . MATH MATH MATH MATH MATH

Finally, we verify ( Existence of orthonormal wavelet bases 1 )-(d) using the proposition ( OST property 2 ). We need to show that MATH we have MATH According to the proposition ( OST property 2 ) it suffices to show that MATH where MATH MATH MATH and MATH We put all together and obtain the requirement MATH where we introduced the convenience notations MATH We simplify the equation: MATH and introduce the functions MATH MATH Thus MATH The requirement MATH may be restated as MATH Therefore, we restate the target of the proof as finding a function MATH such that MATH We rewrite the above relationships in matrix form: MATH and substitute the definition of $m_{1}$ from the proposition ( Scaling equation 2 ): MATH We introduce convenience notations MATH and MATH . We have MATH MATH MATH and we utilize the proposition ( Scaling equation 3 ), MATH , MATH We multiply $\left( \#\right) $ by $M^{\ast}$ and obtain an equivalent equation MATH This proves existence of needed MATH .





Notation. Index. Contents.


















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