Hardy's Theorem and the Short-Time Fourier Transform of Schwartz Functions

Abstract

The Schwartz space of rapidly decaying test functions is characterized by the decay of the short‐time Fourier transform or cross‐Wigner distribution. Then a version of Hardy's theorem is proved for the short‐time Fourier transform and for the Wigner distribution.

Bilinear Localization Operators on Modulation Spaces

“…We restrict ourselves to ( , ) = ⟨ ⟩ ⟨ ⟩ , , ∈ R, since the convolution and multiplication estimates which will be used later on are formulated in terms of weighted spaces with such polynomial weights. As already mentioned, weights of exponential type growth are used in the study of GelfandShilov spaces and their duals in [16,[29][30][31]. We refer to [36] for a survey on the most important types of weights commonly used in time-frequency analysis.…”

“…We refer to [20,21,[29][30][31] for the proof and more details on STFT in other spaces of Gelfand-Shilov type.…”

On the Duality of Regular and Local Functions

“…It is true for the Gaussian Ω(t) ≡ e −πt 2 and herewith expresses Hardy's uncertainty principle [69]. However, it is also true for the Hyperbolic secant Ω(t) ≡ 2/(e t + e −t ), see e.g., [11], and for every fourth Hermite function H, i.e., all H satisfying F H ≡ H. A connection between Gaussians and Hyperbolic Secants is that both belong to a class of "Pólya frequency functions" [70,71].…”

Quantum States and Hardy’s Formulation of the Uncertainty Principle: a Symplectic Approach

“…Finally we wish to mention that Gröchenig and Zimmermann have obtained, in their discussion [11] of uncertainty principles for time-frequency representations, similar results by completely different methods for the shorttime Fourier transform which is just the matrix coefficient of the Heisenberg group. Let f, g ∈ L 2 (R n ).…”