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

TL;DR

This paper investigates the properties and non-triviality conditions of Gelfand-Shilov spaces, providing growth estimates, characterizations via short-time Fourier transforms, and applying these results to analyze the continuity of Toeplitz operators.

Contribution

It establishes the non-triviality condition for Gelfand-Shilov spaces and characterizes these spaces using short-time Fourier transforms, with applications to Toeplitz operator continuity.

Findings

${ m extSigma}^{ extsigma}_s$ is nontrivial iff $s+{ extsigma}>1$

Provides growth estimates for functions and Fourier transforms in these spaces

Characterizes spaces via short-time Fourier transform estimates

Abstract

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