I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 V. Stochastic optimization in one dimension.
 VI. Scalar product in N-dimensions.
 1 Two-sided area of integration, positive a.
 2 Two-sided area of integration, negative a.
 3 Indexing integration domains.
 4 Summary. Calculation of scalar product in N dimensions.
 5 Indexing integration domains II.
 6 Scalar product in N-dimensions. Test case 1.
 7 Scalar product in N-dimensions. Test case 2.
 8 Scalar product in N-dimensions. Test case 3.
 9 Implementation of scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Scalar product in N-dimensions.

e are interested in decomposition of a payoff function , with respect to a tensor product of a wavelet basis . The function has the form for a given vector and and for given parameters .

We introduce the notations

In order to perform decompositions with respect to basis we need to evaluate scalar products of the form . Hence, we proceed with evaluation of the integral

We introduce the index sets and the dimensionality parameter : We assume without loss of generality that We have where The may be evaluated with already developed means, see the section ( Piecewise polynomials in parallel ).

The set has the following representation: We introduce an alternative notation for : where For every dimension we have a subdivision of the interval of integration along the variable . The entire -dimensional domain of the integral has subdivision The restriction is a multidimensional polynomial for each .

For every subdomain that does not contain the boundary the integral decomposes into a product of integrals along every dimension. Such calculation is covered in the section ( Piecewise polynomials ).

We now calculate the integral for a subdomain that intersects the boundary . We use the convenience notation . If then for some numbers , . We arrive to the following representation: Note that the integrals are of the same form as the original integral but in dimensions. However, in order to arrive to a recursive recipe, we need to consider a slightly more complicated area of integration .

 1 Two-sided area of integration, positive a.
 2 Two-sided area of integration, negative a.
 3 Indexing integration domains.
 4 Summary. Calculation of scalar product in N dimensions.
 5 Indexing integration domains II.
 6 Scalar product in N-dimensions. Test case 1.
 7 Scalar product in N-dimensions. Test case 2.
 8 Scalar product in N-dimensions. Test case 3.
 9 Implementation of scalar product in N-dimensions.