Integrating the Wigner Distribution on subsets of the phase space, a Survey

TL;DR

This survey reviews properties of the Wigner distribution integrals over phase space subsets, proving Flandrin's conjecture invalid, and explores mathematical and geometric aspects including boundary cases and integrability conditions.

Contribution

It provides a theoretical proof of Flandrin's conjecture's invalidity, analyzes boundary cases with conic curves, and characterizes the integrability of Wigner distributions using the Feichtinger algebra.

Findings

Flandrin's conjecture is false, proven theoretically.

Wigner distribution of generic L2 pulses is not integrable.

Integrals over convex polygons depend weakly on the number of vertices.

Abstract

We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [MR4054880] with B.Delourme and T.Duyckaerts. We use also the J.G.Wood and A.J.Bracken paper [MR2131219], for which we offer a mathematical perspective. We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper [MR2761287] by E.Lieb and Y.Ostrover. Using the Feichtinger algebra, we show that, generically in the Baire sense, the Wigner distribution of a pulse in $L^2(\mathbb R^n)$ does not belong to $L^1(\mathbb R^{2n})$, providing as a byproduct a large class of examples of subsets of the phase space $\mathbb R^{2n}$ on which the integral of the Wigner distribution is infinite. We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.