We introduce new subclasses of Fourier hyperfunctions of mixedtype, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of theseasymptoticandtemperedhyperfunctions to knownclasses of test functions and distributions, especially the Gel'fand-Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than anynegative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher-dimensional case by applying theRadon transformationfor hyperfunctions.
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