  We introduce new subclasses of Fourier hyperfunctions of mixed type,
satisfying polynomial growth conditions at infinity, and develop their sheaf
and duality theory. We use Fourier transformation and duality to examine
relations of these 'asymptotic' and 'tempered' hyperfunctions to known classes
of test functions and distributions, especially the Gelfand-Shilov-Spaces.
Further it is shown that the asymptotic hyperfunctions, which decay faster than
any negative power, are precisely the class that allow asymptotic expansions at
infinity. These asymptotic expansions are carried over to the
higher-dimensional case by applying the Radon transformation for
hyperfunctions.
