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 Ss , Ss , S-s and S-s. 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 oper-ators are continuous on the one-parameter spaces. Lastly, it is determined that S-s(s) is nontrivial if and only if s + s > 1.