21,595 research outputs found
Fourier transformation of Sato's hyperfunctions
A new generalized function space in which all Gelfand-Shilov classes
() 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 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.Comment: 34 pages, final version, accepted for publication in Adv. Mat
Localization properties of highly singular generalized functions
We study the localization properties of generalized functions defined on a
broad class of spaces of entire analytic test functions. This class, which
includes all Gelfand--Shilov spaces with ,
provides a convenient language for describing quantum fields with a highly
singular infrared behavior. We show that the carrier cone notion, which
replaces the support notion, can be correctly defined for the considered
analytic functionals. In particular, we prove that each functional has a
uniquely determined minimal carrier cone.Comment: 12 pages, published versio
- …