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

## Two-sided area of integration, positive a. e continue derivations of the previous section for the area of integration . We introduce the following convenience notation: If and then . We calculate for , : Let then There are six possibilities of how two intervals might locate relatively to each other: We consider each possibility and express it in terms of at most 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:     The recursion ends with : where one of the numbers , may be .

It remains to calculate the numbers .

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