Asymptotic Hyperfunctions, Tempered Hyperfunctions, and Asymptotic Expansions

Abstract

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 Gelfand-Shilov-Spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any negative power, are precisely the class that allow asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher-dimensional case by applying the Radon transformation for hyperfunctions.

31 pages, 1 figure, typos corrected, references added

References in corpus (5)

• On summability of distributions and spectral geometry
• An Extension of Distribution Theory Related to Gauge Field Theory
• Infinite Infrared Regularization and a State Space for the Heisenberg Algebra
• Mathematical Problems of Gauge Quantum Field Theory: A Survey on the Schwinger Model
• Euclidean Reconstruction in Quantum Field Theory: Between tempered distributions and Fourier Hyperfunctions

Cited by in corpus (7)

• A Note on Fourier-Laplace Transform and Analytic Wave front Set in Theory of Tempered Ultrahyperfunctions
• Mathematics of the Quantum Zeno Effect
• The Edge of the Wedge Theorem for Tempered Ultrahyperfunctions
• The Edge of the Wedge Theorem for Tempered Ultrahyperfunctions II. A Generalized Version
• Structural theorems for quasiasymptotics of ultradistributions
• A Uniqueness Theorem and Its Application to Field-Theoretical Models with a Fundamental Length
• Holomorphic Extension Theorem for Tempered Ultrahyperfunctions