  We elucidate the relation between the two ways of formulating causality in
nonlocal quantum field theory: using analytic test functions belonging to the
space $S^0$ (which is the Fourier transform of the Schwartz space $\mathcal D$)
and using test functions in the Gelfand-Shilov spaces $S^0_\alpha$. We prove
that every functional defined on $S^0$ has the same carrier cones as its
restrictions to the smaller spaces $S^0_\alpha$. As an application of this
result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular
generalized functions of tempered growth and obtain the corresponding extension
of Vladimirov's algebra of functions holomorphic on a tubular domain.
