Almost periodic hyperfunctions

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.