# On integrals over a convex set of the Wigner distribution

## TL;DR

This paper constructs a specific example of a function whose Wigner distribution's integral over a square exceeds 1, answering a question from 1988 and involving Weyl quantization and numerical analysis.

## Contribution

It provides the first counterexample to a longstanding question about the integrals of Wigner distributions over convex sets.

## Key findings

- Wigner distribution integral can exceed 1 over certain convex sets.
- Counterexample disproves previous assumptions about Wigner distribution bounds.
- Numerical analysis supports the theoretical construction.

## 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.