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Quantitative Analysis
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Numerical Analysis
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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.
3. Finite element method.
4. Construction of approximation spaces.
A. Finite element.
B. Averaged Taylor polynomial.
C. Stable space splittings.
D. Frames.
E. Tensor product splitting.
F. Sparse tensor product. Cure for curse of dimensionality.
a. Definition of sparse tensor product.
b. Wavelet estimates in Sobolev spaces.
c. Stability of wavelet splitting.
d. Stable splitting for tensor product of Sobolev spaces.
e. Approximation by sparse tensor product.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Stability of wavelet splitting.


roposition

(Stability of wavelet splitting 1) Assume the condition ( Sparse tensor product setup ). Then for any MATH , $s<r$ and a decomposition MATH we have MATH for $s<r$ . The first inequality holds for $s\leq r$ .

Remark

The above formulation is suboptimal. The result actually takes place for $s\leq r$ . The original sources contain a generic result. However, such sources are out of print and out of stock in online book stores. The below proof is a quick fix for the present special case.

Proof

We estimate MATH and apply the proposition ( Bernstein inequality for wavelets ), recalling MATH MATH Therefore, we obtain the first part of inequality $\left( \#\right) $ . To obtain the second part of $\left( \#\right) $ we apply the proposition ( Jackson inequality for wavelets 2 ): MATH We would like to conclude MATH To prove such inequality we assume the alternative and find a contradiction. The contrary statement is the following: there exists a sequence MATH s.t. MATH and MATH Note that MATH where the $\psi$ has compact support. Hence, even in MATH only a finite number of terms $\psi_{d,k}$ is not orthogonal and such finite number is independent of $d$ . Thus, it suffices to disprove MATH We take a subsequence of MATH converging to a weak limit $f$ (see the proposition ( Weak compactness of bounded set )) and arrive to the task of disproving existence of MATH such that the decomposition MATH has divergent series MATH The sum $\sum_{k}$ is finite and has MATH terms.

According to the proposition ( Bernstein inequality for wavelets ) MATH and according to the formulas ( Property of scale and transport 2 ) and ( Property of scale and transport 3 ) MATH The estimate for the terms MATH was established in the proposition ( Vanishing moments vs approximation ): MATH Therefore MATH and we have a convergence guarantee for the series $S$ for $s<r$ .





Notation. Index. Contents.


















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