Gabor frames and asymptotic behavior of Schwartz distributions

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.Comment: 14 pages. Results extended to the multidimensional cas

Schatten properties, nuclearity and minimality of phase shift invariant spaces

“…We also remark that by [30] it follows that the canonical dual window of an element in Σ 1 (R d ) \ 0 belongs to Σ 1 (R d ). In fact, let K 1 (R d ) be as in [30], then it is clear that K [8].…”

“…We also remark that by [30] it follows that the canonical dual window of an element in Σ 1 (R d ) \ 0 belongs to Σ 1 (R d ). In fact, let K 1 (R d ) be as in [30], then it is clear that K [8]. By [30] it follows that we may choose both φ 1 , φ 2 and their dual windows in Σ 1 (R d ).…”

“…More refined, by [30] we may choose both φ 1 , φ 2 and their dual windows to belong to Σ 1 (R d ) (cf. Remark 1.5).…”

“…In fact, let K 1 (R d ) be as in [30], then it is clear that K [8]. By [30] it follows that we may choose both φ 1 , φ 2 and their dual windows in Σ 1 (R d ).…”