I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 1 Decomposition of payoff function in one dimension. Adaptive multiscaled approximation.
 2 Constructing wavelet basis with Dirichlet boundary conditions.
 3 Accelerated calculation of Gram matrix.
 4 Adapting wavelet basis to arbitrary interval.
 5 Solving one dimensional elliptic PDEs.
 6 Discontinuous Galerkin technique II.
 7 Solving one dimensional Black PDE.
 8 Solving one dimensional mean reverting equation.
 V. Stochastic optimization in one dimension.
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Accelerated calculation of Gram matrix.

e will be repeatedly calculating matrixes of the form for some fixed functions and and a scalar product in . We use properties of scale and transport operations to reduce the number scalar product evaluations.

Let . We calculate and use the formula ( Property of scale and transport 2 ). We use the formula ( Property of scale and transport 3 ). We use the formula ( Property of scale and transport 7 ). Therefore, we regard the scalar product as a function of two parameters and . Together with the rule this significantly reduces the amount of calculation.

In case of a wavelet basis, adapted to an interval , we can apply this rule to the internal functions.