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

Abstract

We study the short-time Fourier transform on the space K-1'(R-n) of distributions of exponential type. We give characterizations of K-1'(R-n) and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform

Gabor frames and asymptotic behavior of Schwartz distributions

“…where C is given by the limit (37). The rest of the proof goes along the same lines as that of [12,Thm. 6.5].…”

“…Our main result is a characterization of the S-asymptotic behavior in terms of the short-time Fourier transform. The authors and Pilipović have recently obtained various Tauberian theorems for short-time Fourier transforms of distributions [12]. We show here in Section 4 that those Tauberian theorems can be considerably improved by discretizing the frequency variable if one employs windows generating a Gabor frame.…”

“…as |h| → ∞ for denoting (27). The next theorem shows that it is possible to improve [12,Thm. 6.2] by discretizing the frequency variable in the STFT if one employs a window ψ that generates a Gabor frame.…”

“…Furthermore, if the action of the timefrequency shifts is continuous on A(R d ), then V ψ f ∈ C(R 2d ). See [5] for the analysis of the STFT for the cases A = L 2 and S; we refer to [12] for a STFT theory based on the spaces K ′ 1 and K 1 . We also mention the interesting paper [1], where window functions are even allowed to be distributions.…”

A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

“…This is stated in[5] under the assumptions pM.1q and pM.2q, but one can relax pM.2q to pM.2q 1 using the continuity result [6, Proposition 1] and adapting the arguments given in[16, Section 3] or[5].…”