On Integrals Over a Convex Set of the Wigner Distribution

  • Bérangère Delourme
  • Thomas Duyckaerts
  • Nicolas LernerEmail author


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 Flandrin (Proc IEEE Int Conf Acoustics 4(1):2176–2179, 1988). Our arguments are based upon the study of the Weyl quantization of the characteristic function of \({{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}}\) along with a precise numerical analysis of its discretization.


Wigner distribution Convexity Weyl quantization 

Mathematics Subject Classification

35P05 81S30 81Q10 



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Authors and Affiliations

  • Bérangère Delourme
    • 1
  • Thomas Duyckaerts
    • 2
  • Nicolas Lerner
    • 3
    Email author
  1. 1.LAGA, UMR 7539, Institut GaliléeUniversité Paris 13VilletaneuseFrance
  2. 2.Institut Universitaire de France & LAGA, UMR 7539, Institut GaliléeUniversité Paris 13VilletaneuseFrance
  3. 3.Institut de Mathématiques de JussieuSorbonne Université (former Paris VI)Paris cedex 05France

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