Abstract. 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) 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.

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

Des conditions spéciales sont accordées aux membres de la SMF.

Suppose that f is a function on R n such that exp(a | · | 2 )f and exp(b | · | 2 )f are bounded, where a, b > 0. Hardy's Uncertainty Principle asserts that if ab > π 2 , then f = 0, while if ab = π 2 , then f = c exp(−a | · | 2 ). In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.Date: May 6, 2008.

No abstract