  We examine properties of Gelfand-Shilov spaces $S_s$, $S^\sigma$,
$S^\sigma_s$, $\Sigma_s$, $\Sigma^\sigma$ and $\Sigma^\sigma_s$. These are
spaces of smooth functions where the functions or their Fourier transforms
admit sub-exponential decay. It is determined that ${\Sigma}^{\sigma}_s$ is
nontrivial if and only if $s+ {\sigma} > 1$. We find growth estimates on
functions and their Fourier transforms in the one-parameter spaces, and we
obtain characterizations in terms of estimates of short-time Fourier transforms
for these spaces and their duals. Additionally, we determine conditions on the
symbols of Toeplitz operators under which the operators are continuous on
one-parameter spaces.
