Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

summary:The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg's uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks' uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with respect to a particular window function cannot be time-frequency concentrated in a subset of the form $S\times \mathcal B(0,b)$ in the time-frequency plane $\mathbb R^d\times \widehat {\mathbb R}^d$. As a side result we generalize a related result of Donoho and Stark on stable recovery of a signal which has been truncated and corrupted by noise