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L2L translation

Suppose that one can expand the function $\xi_j$ as follows:
 
$\displaystyle \xi_j(\overrightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x ...
...})
\chi_i^j(\overrightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }_1}),$     (2.11)

where $\chi_i^j$ is a certain coefficient of the expansion.

Substituting (2.11) into (2.9), one obtains

\begin{eqnarray*}\int_{S_y} K(\mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }) \va...
... x $ }_1\mbox{\boldmath$ x $ }}) L_i(\mbox{\boldmath$ x $ }_1).
\end{eqnarray*}


Thus the following formula is obtained
 
$\displaystyle L_i(\mbox{\boldmath$ x $ }_1)=\sum_{l} L_l(\mbox{\boldmath$ x $ }_0) \chi_i^l(\overrightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }_1}).$     (2.12)

The formula (2.12) is used to translate the coefficients of the local expansion as the centre of the local expansion is shifted from $\mbox{\boldmath$\space x $ }_0$ to $\mbox{\boldmath$\space x $ }_1$ (See Fig.2.5) and this translation is called ``L2L (Local expansion to(2) Local expansion) translation''.



Ken-ichi Yoshida
2001-07-28