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 | References | Similar Articles | Additional Information

Abstract: We characterize the almost periodic hyperfunctions by showing that the following statements are equivalent for any bounded hyperfunction . (i) is almost periodic. (ii) for every . (iii) There are two functions and an infinite order differential operator such that (iv) The Gauss transform of is almost periodic for every . Here is the space of almost periodic continuous functions, is the Sato space of test functions for the Fourier hyperfunctions, and 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.

**[C1]**I. Cioranescu,*On the abstract Cauchy problem in spaces of almost periodic distributions*, J. Math. Anal. Appl**148**(1990), 440-462. MR**91d:34065****[C2]**-,*The characterization of the almost periodic ultradistributions of Beurling type*, Proc. Amer. Math. Soc**116**(1992), 127-134. MR**92k:46063****[CK]**S.-Y. Chung and D. Kim,*Representation of quasianalytic ultradistributions*, Ark. Mat.**31**(1993), 51-60. MR**94e:46069****[CKL]**S.-Y. Chung, D. Kim and E. G. Lee,*Periodic hyperfunctions and Fourier series*, Proc. Amer. Math. Soc.**128**(2000), 2421-2430. CMP**99:05****[H]**L. Hörmander,*The analysis of linear partial differential operators*I, Springer-Verlag, Berlin-New York, 1983. MR**85g:35002a****[K]**H. Komatsu,*Ultradistributions.*I, J. Fac. Sci. Univ. Tokyo. Sect. IA Math**20**(1973), 25-105. MR**47:9277****[KCK]**K. H. Kim, S.-Y. Chung and D. Kim,*Fourier hyperfunctions as the boundary values of smooth solutions of heat equations*, Publ. RIMS, Kyoto Univ**29**(1993), 289-300. MR**94m:46076****[M]**T. Matsuzawa,*A calculus approach to hyperfunctions*II, Trans. Amer. Math. Soc**313**(1990), 619-654. MR**90g:46062****[S]**L. Schwartz,*Théorie des distributions*, Hermann, Paris, 1966. MR**35:730****[W]**D. V. Widder,*The heat equation*, Academic Press, New York, 1975. MR**57:6840**

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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