Fourier Characterizations and Non-triviality of Gelfand–Shilov Spaces, with Applications to Toeplitz Operators

We find growth estimates on functions and their Fourier transforms in the one-parameter Gelfand–Shilov spaces S s , S σ , Σ s and Σ σ . We obtain characterizations for these spaces and their duals in terms of estimates of short-time Fourier transforms. We determine conditions on the symbols of Toeplitz operators under which the operators are continuous on the one-parameter spaces. Lastly, it is determined that Σ s σ is nontrivial if and only if s + σ > 1 .