  A new generalized function space in which all Gelfand-Shilov classes
$S^{\prime 0}_\alpha$ ($\alpha>1$) of analytic functionals are embedded is
introduced. This space of {\it ultrafunctionals} does not possess a natural
nontrivial topology and cannot be obtained via duality from any test function
space. A canonical isomorphism between the spaces of hyperfunctions and
ultrafunctionals on $R^k$ is constructed that extends the Fourier
transformation of Roumieu-type ultradistributions and is naturally interpreted
as the Fourier transformation of hyperfunctions. The notion of carrier cone
that replaces the notion of support of a generalized function for
ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing
the Laplace transformation of ultrafunctionals carried by proper convex closed
cones is obtained and the connection between the Laplace and Fourier
transformation is established.
