On Integrals Over a Convex Set of the Wigner Distribution

摘要

We provide an example of a normalized L-2( R) function u such that its Wigner distribution W(u, u) has an integral > 1 on the square [0, a] x [0, a] for a suitable choice of a. This provides a negative answer to a question raised by Flandrin (Proc IEEE Int Conf Acoustics 4(1):2176-2179, 1988). Our arguments are based upon the study of the Weyl quantization of the characteristic function of R+ x R+ along with a precise numerical analysis of its discretization.