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

By Karlheinz Gröchenig, G. Zimmermann. 2001

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Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms (2001, background): “Zimmermann in a forthcoming paper ([15]).”
Time–Frequency analysis of localization operators (2003, unknown): “For instance, they both map L2ðRÞ L2ðRÞ into L2ðRÞ andSðRÞ SðRÞ intoSðRÞ: Furthermore, they can be extended to a map from S0ðRdÞ S0ðRdÞ into S0ðR2dÞ: We first list some crucial properties of the STFT (for proofs, see [23, Chapter 3] and [25]).”
Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class (2006, methodology): “We will also make use of the following formula concerning the Fourier transform of a product of STFTs, which follows from an easy computation (carried out in [32] and in [21]) (Vg1f1Vg2f2) ∧ (ξ, x) = (Vf2f1Vg2g1)(−x, ξ) .”