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

Abstract

AbstractWe find growth estimates on functions and their Fourier transforms in the one-parameter Gelfand–Shilov spaces $$S_s$$ S s , $$S^\sigma $$ S σ , $$\Sigma _s$$ Σ s and $$\Sigma ^\sigma $$ 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 $$\Sigma _s^\sigma $$ Σ s σ is nontrivial if and only if $$s+\sigma > 1$$ s + σ > 1 .