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

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.