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Periodic hyperfunctions and Fourier series


Authors: Soon-Yeong Chung, Dohan Kim and Eun Gu Lee
Journal: Proc. Amer. Math. Soc. 128 (2000), 2421-2430
MSC (1991): Primary 46F15, 35K05, 42B05
DOI: https://doi.org/10.1090/S0002-9939-99-05281-8
Published electronically: December 7, 1999
MathSciNet review: 1657782
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Abstract: Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients $c_{\alpha }$ of periodic hyperfunctions are of infra-exponential growth in $\mathbb{R}^{n}$, i.e., $c_{\alpha }< C_{\epsilon }e^{\epsilon |\alpha |}$ for every $\epsilon >0$ and every $\alpha \in \mathbb{Z}^{n}$. This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions.

To show these we introduce the space $\mathcal{B}_{L^{p}}$ of hyperfunctions of $L^{p}$ growth which generalizes the space $\mathcal{D}'_{L^{p}}$ of distributions of $L^{p}$ growth and represent generalized functions as the initial values of smooth solutions of the heat equation.


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Additional Information

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: dohankim@snu.ac.kr

Eun Gu Lee
Affiliation: Department of Mathematics, Dongyang Technical College, Seoul 152–714, Korea
Email: eglee@orient.dytc.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-99-05281-8
Keywords: Hyperfunction, periodic, Fourier series
Received by editor(s): June 16, 1998
Received by editor(s) in revised form: September 24, 1998
Published electronically: December 7, 1999
Additional Notes: Partially supported by BSRI and GARC–KOSEF
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society