On Integrals Over a Convex Set of the Wigner Distribution

Abstract

We provide an example of a normalized $$L^{2}({\mathbb {R}})$$ function u such that its Wigner distribution $${\mathcal {W}}(u,u)$$ has an integral $$>1$$ on the square $$[0,a]\times [0,a]$$ 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 $${{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}}$$ along with a precise numerical analysis of its discretization.