  We give some precisions on the Fourier-Laplace transform theorem for tempered
ultrahyperfunctions introduced by Sebasti\~ao e Silva and Hasumi, by
considering the theorem in its simplest form: the equivalence between support
properties of a distribution in a closed convex cone and the holomorphy of its
Fourier-Laplace transform in a suitable tube with conical basis. We establish a
generalization of Paley-Wiener-Schwartz theorem for this setting. This theorem
is interesting in connection with the microlocal analysis, where a description
of the singularity structure of tempered ultrahyperfunctions in terms of the
concept of analytic wave front set is given. We also suggest a physical
application of the results obtained in the construction and study of field
theories with fundamental length.
