Fourier transformation of Sato's hyperfunctions
Abstract
A new generalized function space in which all GelfandShilov 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 Roumieutype 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 PaleyWienerSchwartztype 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2004
 arXiv:
 arXiv:math/0401151
 Bibcode:
 2004math......1151S
 Keywords:

 Functional Analysis;
 Complex Variables;
 46F15;
 32A45
 EPrint:
 34 pages, final version, accepted for publication in Adv. Math