Tensor product based precomputation

Using the fact that the window functions are built as tensor products one can store $\varphi_t((\ensuremath{\boldsymbol{x}}_j)_t - \frac{l_t}{n_t})$ for $j=0,\hdots,M-1$ , $t=0,\hdots,d-1$ , and $l_t\in I_{n_t,m} ((\ensuremath{\boldsymbol{x}}_j)_t)$ where $(\ensuremath{\boldsymbol{x}}_j)_t$ denotes the $t$ -th component of the $j$ -th node. This method uses a medium amount of memory to store $d(2m+1)M$ real numbers in total. However, one has to carry out for each node at most $2(2m+1)^d$ extra multiplications to compute from the factors the multivariate window function $\varphi(\ensuremath{\boldsymbol{x}}_j - \ensuremath{\boldsymbol{n}}^{-1}\odot\ensuremath{\boldsymbol{l}})$ for $\ensuremath{\boldsymbol{l}}\in I_{\ensuremath{\boldsymbol{n}},m} (\ensuremath{\boldsymbol{x}}_j)$ . Note, that this technique is available for every window function discussed here and can be used by means of the flag PRE_PSI which is also the default method within our software library.