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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.
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Two-sided area of integration, negative a.


e repeat calculations of the previous sections for $a_{n-1}<0$ . We are still using the notations MATH , MATH , MATH . The area of integration is still MATH . Hence MATH We calculate for $a_{n-1}<0$ : MATH Let MATH then MATH We consider the following cases MATH For each possibility we derive a two-sided inequality for MATH : MATH MATH MATH MATH MATH MATH We now express the value of MATH for each of these six cases: MATH for some numbers MATH , $\tilde{C}_{n-1}$ . We arrive to the following recursive representation: MATH MATH MATH MATH MATH

We proceed to calculate the numbers MATH .




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

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