Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where certain frequencies occur in the input signal, this method was introduced by Dennis Gabor. In this paper, we generalize the classical Gabor–Fourier transform (GFT) to the Riemannian symmetric space calling it the Helgason–Gabor–Fourier transform (HGFT). We prove several important properties of HGFT like the reconstruction formula, the Plancherel formula and Parseval formula. Finally, we establish some local uncertainty principle such as Benedicks-type uncertainty principle.
Helgason–Gabor–Fourier transform and uncertainty principles