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.

## Sparse tensor product. Cure for curse of dimensionality.

e build on results of the section ( Elementary definitions of wavelet analysis ).

Condition

(Sparse tensor product setup) We introduce the mesh (see the section Wavelet analysis ) on . We assume that and are dual GMRAs, scaling functions and wavelets on , see the sections ( Compactly supported smooth biorthogonal wavelets ) and ( Construction of MRA and wavelets on half line or an interval ). We assume that satisfies condition ( Frame formula 1 ) of the proposition ( Frame property 2 ) and we showed how to achieve that in the section ( Spline functions ).

Note that we denote the entire -basis of semi-orthogonal functions constructed in the section ( Construction of MRA and wavelets on half line or an interval ). In particular, the statement of the proposition ( Reproduction of polynomials 4 ) adapted to GMRA would read for some , (thus, there are vanishing moments).

We keep the notation but it now includes the space so that is in the closure of the linear span of with being what is in the notation of the definitions ( Generalized multiresolution analysis ) and ( Dual GMRA ).

Therefore, the index in the notation refers to the inclusion rather than scale operation and the index refers to the numbering of functions produced by the procedure of the section ( Construction of MRA and wavelets on half line or an interval ) rather than shift operation. However, the old and new meaning of these indexes is close, in particular, there is -norm stability when changing these indexes, see the formulas ( Property of scale and transport 2 ),( Property of scale and transport 3 ).

We keep the notation for the spaces with the property

Moreover, for some .

We use the notation which is finite because we restrict our attention to bounded sets. Furthermore,

We will be extending wavelet approximation to the domain .

 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.
 Notation. Index. Contents.