Fourier characterizations and non-triviality of Gelfand-Shilov spaces, with applications to Toeplitz operators

Abstract

We examine properties of Gelfand-Shilov spaces $S_s$, $S^{\sigma }$, $S_s^{\sigma }$, ${\Sigma }_s$, ${\Sigma }^{\sigma }$ and ${\Sigma }_s^{\sigma }$. These are spaces of smooth functions where the functions or their Fourier transforms admit sub-exponential decay. It is determined that ${\Sigma }_s^{\sigma }$ 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.

18 pages. This is the first version and it is expected that there will be some changes in future versions

Cited by in corpus (1)

• Global hypoellipticity on time-periodic Gelfand-Shilov spaces via non-discrete Fourier analysis