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.

## Summary. Calculation of scalar product in N dimensions.

e start from a set and a function : for a given vector and and We introduce the notation for the set of scales in the definition of , the notation for the set of translations, the notation for the intervals of integration and the notation for multi-indexes.

We fix and concentrate on calculating where

1. Form

2. Let be a one-to-one onto mapping: for some . For every we form the subdomain and the set of vertices of . For each pair we calculate

Let be a one-to-one onto mapping: for some , where Let be a one-to-one onto mapping: for some , where Thus indexes boundary subdomains and indexes internal subdomains.

3. We introduce

4. For every we form and calculate

5. For every we form and calculate as follows: where we adopt the convention and calculate recursively in as follows.

For , For , , , where For , , , For , , ,

For , , , where For , , , For , , ,