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(2011 / vol.13 / no.1)
HEAT KERNEL METHOD AND FOURIER ANALYSIS ON HYPERFUNCTIONS
DOHAN KIM
Pages. 15-34   



The Sato theory of hyperfunctions and Fourier hyperfunctions is a really natural extension of the Schwartz theory of distributions and tempered distributions. We show that the naturalness of hyperfunctions comparing our results in hyperfunctions and the corresponding results in distributions in such areas as the characterization of test function spaces for Fourier transformation, both periodic and almost periodic distributions and hyperfunctions and Bochner-Schwartz theorem for (conditionally) positive definite distributions and hyperfunctions.
We have one important guiding theme to compare distributions and
hyperfunctions. Estimates related to distributions are {it
tempered} or {it polynomially increasing} and estimates related to
hyperfunctions are of {it infra-exponential growth}.
Also, in order to define periodicity, (conditionally) positive definiteness for the hyperfunctions we make use of Matsuzawa's heat kernel method which represents various generalized functions including distributions and hyperfunctions as the limits of initial values of the solutions of the heat equation satisfying suitable
growth conditions.



1. Introduction
2. Characterization of various test functions spaces via Fourier transform
3. Generalized functions as initial values of solutions of the heat equation
4. Periodic hyperfunctions and distributions
5. Almost periodic hyperfunctions and distributions
6. Positive definite hyperfunctions and distributions
7. Conditionally positive definite (Fourier) hyperfunctions and distributions