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

2022/02/02 by Petersson, Albin · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2202.00938

Abstract

We examine properties of Gelfand-Shilov spaces S_s, S^σ, S^σ_s, Σ_s, Σ^σ and Σ^σ_s. These are spaces of smooth functions where the functions or their Fourier transforms admit sub-exponential decay. It is determined that Σ^σ_s is nontrivial if and only if s+ σ > 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.

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