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Periodic hyperfunctions and Fourier series

Authors:	Soon-Yeong Chung Dohan Kim Eun Gu Lee

Institution:	Department of Mathematics, Sogang University, Seoul 121--742, Korea ; Department of Mathematics, Seoul National University, Seoul 151--742, Korea ; Department of Mathematics, Dongyang Technical College, Seoul 152--714, Korea

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.

Keywords:	Hyperfunction periodic Fourier series

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