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.
 A. Calculation of w1 for negative a.
 B. Calculation of w2 for negative a.
 C. Calculation of w3 for 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.

## Two-sided area of integration, negative a. e repeat calculations of the previous sections for . We are still using the notations , , . The area of integration is still . Hence We calculate for : Let then We consider the following cases For each possibility we derive a two-sided inequality for :      We now express the value of for each of these six cases: for some numbers , . We arrive to the following recursive representation:     We proceed to calculate the numbers .

 A. Calculation of w1 for negative a.
 B. Calculation of w2 for negative a.
 C. Calculation of w3 for negative a.