  In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized
function space $\mathcal U(R^k)$ which can be naturally interpreted as the
Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown
that all Gelfand--Shilov spaces $S^{\prime 0}_\alpha(R^k)$ ($\alpha>1$) of
analytic functionals are canonically embedded in $\mathcal U(R^k)$. While the
usual definition of support of a generalized function is inapplicable to
elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$, their
localization properties can be consistently described using the concept of {\it
carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm.
Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier
cones of elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$ is
studied. It is proved that an analytic functional $u\in S^{\prime
0}_\alpha(R^k)$ is carried by a cone $K\subset R^k$ if and only if its
canonical image in $\mathcal U(R^k)$ is carried by $K$.
