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

Approximation by sparse tensor product.


roposition

(Sparse tensor product approximation 1) Assume the condition ( Sparse tensor product setup ). Then MATH

Proof

First, we verify that MATH where MATH stands for integer part. Indeed, according to the definition ( Sparse tensor product ), MATH and MATH

Clearly MATH

Hence, the inclusion $\left( \#\right) $ is proven.

Next, we derive the statement from the propositions ( Jackson inequality for wavelets ), ( Tensor product of function spaces ) and ( Sobolev spaces in N dim as tensor products ). MATH We use the proposition ( Sobolev spaces in N dim as tensor products ), the MATH has representations MATH for various placements of $g_{k}$ among $u_{k}$ .

We continue estimation MATH We use the proposition ( Jackson inequality for wavelets ) for each term: MATH Hence MATH We use the proposition ( Sobolev spaces in N dim as tensor products ). MATH

Proposition

(Sparse tensor product approximation 2) Assume the condition ( Sparse tensor product setup ). Then for MATH , MATH $0\leq s_{k}\leq n$ , $k=1,...,N$ we have MATH

Proof

The proof is essentially the same as the proof of the proposition ( Sparse tensor product approximation 1 ). We estimate MATH We use the definition ( Sobolev spaces with dominating mixed derivative ), the MATH has representation MATH

We continue estimation MATH We use the proposition ( Jackson inequality for wavelets ) for each term: MATH





Notation. Index. Contents.


















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