Integrating The Wigner Distribution On Subsets Of The Phase Space A Survey Nicolas Lerner by Nicolas Lerner 9783985475711, 9783985470716, 3985475717, 3985470715 instant download after payment.

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 [J. Fourier Anal. Appl. 26 (2020), no. 1, article no. 6] with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [J. Math. Phys. 46 (2005), no. 4, article no. 042103], 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 [J. Math. Phys. 51 (2010), no. 10, article no. 102101] 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  providing as a byproduct a large class of examples of subsets of the phase space R 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.