  We extend an uncertainty principle due to Beurling into a characterization of
Hermite functions. More precisely, all functions $f$ on $\R^d$ which may be
written as $P(x)\exp (Ax,x)$, with $A$ a real symmetric definite positive
matrix, are characterized by integrability conditions on the product
$f(x)\hat{f}(y)$. We also give the best constant in uncertainty principles of
Gelf'and Shilov type. 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.
