Periodic hyperfunctions and Fourier series

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.