Fourier transformation of Sato's hyperfunctions

2004/01/31 by A. G. Smirnov
Mathematics · #math.FA #math.CV #msc:46F15 #msc:32A45

paper · pdf

published as Adv. Math. 196 (2005) 310-345 · 34 pages, final version, accepted for publication in Adv. Math

arxiv created 2004/09/30 · arxiv updated 2009/12/01

Abstract

A new generalized function space in which all Gelfand-Shilov classes S^{′ 0}_α (α>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.

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