Gabor frames and asymptotic behavior of Schwartz distributions

TL;DR

This paper characterizes the asymptotic behavior of Schwartz distributions using Gabor frames, establishing Tauberian theorems for shift asymptotics via short-time Fourier transforms.

Contribution

It introduces new characterizations of distribution asymptotics through Gabor frame-based Tauberian theorems, linking distribution spaces via Gabor coefficient operators.

Findings

Gabor coefficient operator provides topological isomorphisms between distribution spaces.

Characterizations of asymptotic properties of Schwartz distributions using Gabor frames.

Establishment of Tauberian theorems for shift asymptotics in distribution analysis.

Abstract

We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions $\mathcal{S}'(\mathbb{R}^d)$ and distributions of exponential type $\mathcal{K}'_{1}(\mathbb{R}^{d})$ onto their images.