INTEGRATING THE WIGNER DISTRIBUTION ON SUBSETS OF THE PHASE SPACE, A SURVEY

Overview & Synopsis

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 [4] with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [37], 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 [27] by E. Lieb and Y. Ostrover. Using the Baire Category Theorem, we show that, generically, 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 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.