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.
Categories: math.FA