Almost periodic hyperfunctions

We characterize the almost periodic hyperfunctions by showing that the foll owing statements are equivalent for any bounded hyperfunction T. (i)T is al most periodic. (ii) T * phi is an element of C-ap for every phi is an eleme nt of F. (iii) There are two functions f, g is an element of C-ap and an in finite order differential operator P such that T = P(D-2)f + g: (iv) The Ga uss 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, F is the S ato 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.