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 P. Flandrin in 1988. Our arguments are based upon the study of the Weyl quantization of the indicatrix of ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ along with a precise numerical analysis of its discretization.

34 pages, 1 figure