We prove that every continuous linear functional on the space $S^0(R^d)$
consisting of the entire analytic functions whose Fourier transforms belong to
the Schwartz space $\mathcal D$ has a unique minimal carrier cone in $R^d$,
which substitutes for the support. The proof is based on a relevant
decomposition theorem for elements of the spaces $S^0(K)$ associated naturally
with closed cones $K\subset R^d$. These results, essential for applications to
nonlocal quantum field theory, are similar to those obtained previously for
functionals on the Gelfand-Shilov spaces $S^0_\alpha$, but their derivation is
more sophisticated because $S^0(K)$ are not DFS spaces and have more
complicated topological structure.