We characterize possible bounds for representing functions of arbitrary hyperfunctions. Specifically, there are always representing functions decreasing rapidly outside each strip near $\mathbb{R}$. Also, exponential decrease of any type on any strip $\mathbb{R}\times \pm i[c,C], 0<c<C<\infty,$ can be achieved. This will be used in [10] to define an asymptotic Fourier and Laplace transformation on the space of hyperfunctions.
Funct. Approx. Comment. Math.
44(1):
33-44
(March 2011).
DOI: 10.7169/facm/1301497745