2021/02/16 by Nicolás Lerner, Lerner, Nicolas · 2 citations
Mathematics · #35P05 #42B20 #47G10 #47G30 #47N70 #94A12 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Random Matrices and Applications #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2102.08090
openalex publication_date 2021/02/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 L2(\mathbb Rn) does not belong to L1(\mathbb R2n), providing as a byproduct a large class of examples of subsets of the phase space \mathbb R2n 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.