Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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

Proof

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

Clearly

Hence, the inclusion 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 ). We use the proposition ( Sobolev spaces in N dim as tensor products ), the has representations for various placements of among .

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

Proposition

(Sparse tensor product approximation 2) Assume the condition ( Sparse tensor product setup ). Then for , , we have

Proof

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

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

 Notation. Index. Contents.