Almost periodic hyperfunctions

Abstract

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 ∗ φ ∈ Cap for every φ ∈ F . (iii) There are two functions f, g ∈ Cap and an infinite order differential operator P such that T = P (D2)f + g. (iv) The Gauss transform u(x, t) = T ∗ E(x, t) of T is almost periodic for every t > 0. Here Cap is the space of almost periodic continuous functions, 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.