  We study function spaces consisting of analytic functions with fast decay on
horizontal strips of the complex plane with respect to a given weight function.
Their duals, so called spaces of (ultra)hyperfunctions of fast growth,
generalize the spaces of Fourier hyperfunctions and Fourier
ultrahyperfunctions. An analytic representation theory for their duals is
developed and applied to characterize the non-triviality of these function
spaces in terms of the growth order of the weight function. In particular, we
show that the Gelfand-Shilov spaces of Beurling type
$\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$
are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} <
\infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study
boundary values of holomorphic functions in spaces of ultradistributions of
exponential type, which may be of quasianalytic type.
