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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.
a. Adapting MRA to the interval [0,1].
b. Adapting wavelets to interval [0,1].
c. Adapting GMRA to interval [0,1].
d. Adapting dual wavelets to interval [0,1].
e. Constructing dual GMRA on [0,1] with boundary conditions.
3. Finite element method.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Adapting MRA to the interval [0,1].


verywhere in this section we assume that the following condition holds. See the proposition ( Existence of smooth compactly supported wavelets ) for context.

Condition

(Compactly supported scaling function) A scaling function $\phi$ comes from an MRA and is compactly supported: MATH Moreover, the associated wavelet $\psi$ satisfies MATH

Therefore, according to the proposition ( Reproduction of polynomials 4 ), MATH where the $\QTR{cal}{P}_{N}$ is the collection of all polynomials with degree up to and including $N$ .

We introduce the notations MATH so that MATH

Condition

(Sufficiently fine scale) We assume that the parameter $d$ is sufficiently large so that MATH

For a polynomial MATH we introduce the notation MATH In particular, MATH and, according to the proposition ( Reproduction of polynomials 4 ), MATH

Definition

([0,1]-adapted MRA) We define the spaces MATH where the sign $\oplus$ refers to the MATH -orthogonality.

Proposition

The spaces $V_{d,L}$ , $V_{d,I}$ and $V_{d,R}$ are mutually MATH -orthogonal.

Proof

The statement follows from construction: the collection MATH is MATH -orthogonal and the spaces $V_{d,L}$ , $V_{d,I}$ and $V_{d,R}$ are linear span of $\phi_{d,k}$ for different values of the index $k$ .

We would like to find an MATH -orthogonal basis for the space MATH . The functions MATH are already MATH -orthogonal. Before we perform Gram-Schmidt orthogonalization of MATH and MATH we need to establish that both collections are linearly independent.

Proposition

(Restricted linear independence) The collection MATH is linearly independent on $\left[ 0,1\right] $ .

Proof

The proof may be found in [Lemarie] or [Meyer] . For a particular $\phi$ this may be verified directly.

Proposition

(Maximal dimension 1) The collection MATH is linearly independent. The collection MATH is linearly independent.

Proof

We introduce the notation MATH We aim to establish that the collection MATH is linearly independent then the statement would follow from the proposition ( Restricted linear independence ).

We have MATH We make the change $y=2^{d}x-k$ , $dx=2^{-d}dy$ , MATH . MATH where MATH and, according to the proposition ( Integral of scaling function ), MATH Thus MATH and linear independence of MATH follows from linear independence of MATH .

We proceed with orthogonalization. We seek the functions MATH such that MATH We substitute the first relationship into the second: MATH The $X$ is a square symmetric positive-definite matrix. Hence, there exists a Choleski decomposition: MATH and the choice MATH is sufficient to produce MATH -orthogonal collection MATH . The collection MATH is MATH -orthogonal to MATH because it is a linear combination of MATH .

Proposition

(Resolution structure for adapted scale functions) The spaces MATH have the structure MATH

Proof

We saw in the proof of the proposition ( Maximal dimension 1 ) that there exist two-way linear transformations MATH where MATH

Due to the conditions MATH is suffices to show that $\varphi_{0,k}$ may be represented as a linear combination of MATH .

We calculate MATH We use the proposition ( Scaling equation ). MATH We use the formula ( Property of scale and transport 7 ). MATH Here $p\in K_{0,L}$ , $r\in\QTR{cal}{Z}$ , hence the index $2p+r$ varies over $\QTR{cal}{Z}$ . We intend to make a change MATH , $q=2p+r$ . Thus $q-r$ can only be even. We separate even and odd values of $r$ . For $r$ even we have $r=2t$ , $q=2s$ , hence, the $q=2p+r$ becomes $s=p+t$ : MATH For $r$ odd we have $r=2t+1$ , $q=2s+1$ , MATH : MATH We make a change $s\rightarrow u$ , $2s=u$ in the first sum and $2s+1=u$ in the second sum and put these sums together because then $u$ would vary over $\QTR{cal}{Z}$ : MATH We have MATH We continue MATH At this point we invoke the proposition ( Sufficient conditions for vanishing moments )-d: MATH We separate even and odd values of $i$ : MATH Note that MATH and $k\in0,...,N-1$ thus the proposition ( Sufficient conditions for vanishing moments )-d applies for all values of $p$ and MATH We continue MATH The expression MATH is within the linear combination of MATH .





Notation. Index. Contents.


















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