Spaces of Test Functions via the STFT

Abstract

We characterize several classes of test functions, among them Björck's ultra-rapidly decaying test functions and the Gelfand-Shilov spaces of typeS, in terms of the decay of their short-time Fourier transform and in terms of their Gabor coefficients.

Ultradistributions and Time-Frequency Analysis

“…Note that Theorem 3.8 for the inductive limit case and the short-time Fourier transform is proved in [25], see also [21 …”

“…For the sake of simplicity we skip the definition of Gelfand-Shilov type spaces S α , S β , and W M × M , and refer the reader to [18], [25], [27]. …”

“…Theorems 3.8 and 3.9 are proved in [25] for the inductive limit case and the short-time Fourier transform. The proof for the Wigner distribution W (f, f ) is given in [8].…”

“…Note that for w s (x, y) For the proof of projective limit case we refer to [39]. The inductive limit case is proved in [25].…”

Localization Operators and Exponential Weights for Modulation Spaces

“…Window functions are assumed to belong to ultra-modulation spaces, cf. [17], [21], [26]; namely, we introduce M 1 ws , with w s exponential weight depending on s, 0 ≤ s < ∞ (the limit case s = 0 gives the unweighted space), having as projective and inductive limit the Gelfand-Shilov space S (1) , S {1} respectively, subspaces of the Schwartz set S. Then, we may allow symbols a which are ultra-distributions in S (1) , and find sufficient and necessary conditions for the L 2 -boundedness. In this way we extend [6], obtaining in particular that for windows ϕ 1 , ϕ 2 ∈ S {1} , every ultra-distribution with compact support a ∈ E t , t > 1 (see [22], [23]), defines a trace class operator A ϕ1,ϕ2 a .…”