  In this paper the generalized localization principle for the spherical
partial sums of the multiple Fourier series in the $L_2$ - class is proved,
that is, if $f\in L_2(T^N)$ and $f=0$ on an open set $\Omega \subset T^N$, then
it is shown that the spherical partial sums of this function converge to zero
almost - everywhere on $\Omega$. It has been previously known that the
generalized localization is not valid in $L_p(T^N)$ when $1\leq p<2$. Thus the
problem of generalized localization for the spherical partial sums is
completely solved in $L_p(T^N)$, $p\geq 1$: if $p\geq2$ then we have the
generalized localization and if $p<2$, then the generalized localization fails.
