The short-time Fourier transform of distributions of exponential type and Tauberian theorems for shift-asymptotics

We study the short-time Fourier transform on the space ${\mathcal{K}}_1^{\mathrm{\prime }}({\mathbb{R}}^n)$ of distributions of exponential type. We give characterizations of ${\mathcal{K}}_1^{\mathrm{\prime }}({\mathbb{R}}^n)$ and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.