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Almost periodic hyperfunctions


Authors: Jaeyoung Chung, Soon-Yeong Chung, Dohan Kim and Hee Jung Kim
Journal: Proc. Amer. Math. Soc. 129 (2001), 731-738
MSC (1991): Primary 46F15, 35K05, 42B05
DOI: https://doi.org/10.1090/S0002-9939-00-05800-7
Published electronically: August 30, 2000
MathSciNet review: 1792186
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Abstract: We characterize the almost periodic hyperfunctions by showing that the following statements are equivalent for any bounded hyperfunction $T$. (i) $T$ is almost periodic. (ii) $T*\varphi \in C_{ap}$ for every $\varphi \in \mathcal{F}$. (iii) There are two functions $f,g \in C_{ap}$ and an infinite order differential operator $P$ such that $T=P(D^{2})f+g.$ (iv) The Gauss transform $u(x,t)=T*E(x,t)$ of $T$ is almost periodic for every $t>0$. Here $C_{ap}$ is the space of almost periodic continuous functions, $\mathcal{F}$ is the Sato space of test functions for the Fourier hyperfunctions, and $E(x,t)$ is the heat kernel. This generalizes the result of Schwartz on almost periodic distributions and that of Cioranescu on almost periodic (non-quasianalytic) ultradistributions to the case of hyperfunctions.


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Additional Information

Jaeyoung Chung
Affiliation: Department of Mathematics, Kunsan National University, Kunsan 573–360, Korea
Email: jychung@ks.kunsan.ac.kr

Soon-Yeong Chung
Affiliation: Department of Mathematics, Sogang University, Seoul 121–742, Korea
Email: sychung@ccs.sogang.ac.kr

Dohan Kim
Affiliation: Department of Mathematics, Seoul National University, Seoul 151–742, Korea
Email: dhkim@math.snu.ac.kr

Hee Jung Kim
Affiliation: Department of Mathematics, Seoul National University, Seoul 151–742, Korea
Email: ciel@math.snu.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-00-05800-7
Keywords: Almost periodic, hyperfunction, ultradistribution
Received by editor(s): May 4, 1999
Published electronically: August 30, 2000
Additional Notes: The first and second authors were partially supported by KOSEF (1999-2-101-001-5). The third and fourth authors were partially supported by BK21
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2000 American Mathematical Society

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