Annihilating sets for the short time Fourier transform (Q981621)

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.