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Annihilating sets for the short time Fourier transform. (English) Zbl 1194.42015
The uncertainty principle means that a non-zero function and its Fourier transform cannot be sharply localized. In this paper, the authors consider support conditions for the short time Fourier transform (STFT). The aim is to obtain a class of subsets in ${\mathbb R}^{2d}$ (called thin sets at infinity) so that the support of the STFT of a signal $f\in L^2({\mathbb R}^d)$ with respect to a non-zero window $g\in L^2({\mathbb R}^d)$ cannot belong to this class unless $f=0$. Moreover it is proved that the $L^2$-norm of the STFT is essentially concentrated in the complement of any thin set at infinity. Finally, this result is generalized to other Hilbert spaces of functions or distributions.

42B10Fourier type transforms, several variables
Full Text: DOI
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