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.

## Stable splitting for tensor product of Sobolev spaces.

efinition

(Sobolev spaces with dominating mixed derivative) For a domain we introduce the classes

See the section ( Tensor product of Hilbert spaces ) for definition of tensor product of Hilbert spaces and associated scalar product.

The above definition is to be compared with the following statement.

Proposition

(Sobolev spaces in N dim as tensor products) For a domain we have

Proof

The spaces on the left and right of have equivalent norms, the intersection of sets corresponds to summation of norms. They also coincide as sets. So see this it suffices to use the proposition ( Tensor product of function spaces ) to construct bases for dense subsets.

Proposition

(Stability of splitting for Sobolev spaces with dominating mixed derivative) Assume the condition ( Sparse tensor product setup ). Then for , , we have for a decomposition

Proof

We calculate the first inequality using the proposition ( Stability of wavelet splitting 1 ): The second inequality is a consequence of the proposition ( Tensor product splitting ).

Proposition

(Stability of splitting for tensor product of Sobolev spaces) Assume the condition ( Sparse tensor product setup ). Then for we have for a decomposition

Proof

The proof is very similar to the proof of the proposition ( Stability of splitting for Sobolev spaces with dominating mixed derivative ).

 Notation. Index. Contents.