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

Gabor frames and asymptotic behavior of Schwartz distributions