International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 5, Pages 755-788
Asymptotic hyperfunctions, tempered hyperfunctions, and asymptotic
Fraunhofer-Institute for Secure Information Technology, Dolivostrasse 15, Darmstadt 64293, Germany
Received 20 September 2004
Copyright © 2005 Andreas U. Schmidt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 Gel'fand-Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any
negative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher-dimensional case by applying the Radon transformation for hyperfunctions.