A note on Fourier–Laplace transform and analytic wave front set in theory of tempered ultrahyperfunctions

In this paper we study the Fourier–Laplace transform of tempered ultrahyperfunctions introduced by Sebastião e Silva and Hasumi. We establish a generalization of Paley–Wiener–Schwartz theorem for this setting. This theorem is interesting in connection with the microlocal analysis. For this reason, the paper also contains a description of the singularity structure of tempered ultrahyperfunctions in terms of the concept of analytic wave front set.