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Integrating the Wigner Distribution on Subsets of the Phase Space, a Survey
 
Nicolas Lerner Sorbonne Université, Paris, France
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-071-6
Product Code:  EMSMEM/12
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
Please note AMS points can not be used for this product
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Integrating the Wigner Distribution on Subsets of the Phase Space, a Survey
Nicolas Lerner Sorbonne Université, Paris, France
A publication of European Mathematical Society
Softcover ISBN:  978-3-98547-071-6
Product Code:  EMSMEM/12
List Price: $75.00
AMS Member Price: $60.00
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Memoirs of the European Mathematical Society
    Volume: 122024; 224 pp
    MSC: Primary 81; Secondary 35; 42; 47; 94

    The author reviews several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, the author provides a theoretical proof of the invalidity of Flandrin’s conjecture, a fact already proven via numerical arguments in his joint paper [J. Fourier Anal. Appl. 26 (2020), no. 1, article no. 6 with B. Delourme and T. Duyckaerts].

    The author also uses the J. G. Wood and A. J. Bracken paper [J. Math. Phys. 46 (2005), no. 4, article no. 042103], for which he offers a mathematical perspective. The author thoroughly reviews the case of subsets of the plane whose boundary is a conic curve and shows 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, the author shows 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. The author also studies 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.

    A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

  • Additional Material
     
     
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    Review Copy – for publishers of book reviews
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Volume: 122024; 224 pp
MSC: Primary 81; Secondary 35; 42; 47; 94

The author reviews several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, the author provides a theoretical proof of the invalidity of Flandrin’s conjecture, a fact already proven via numerical arguments in his joint paper [J. Fourier Anal. Appl. 26 (2020), no. 1, article no. 6 with B. Delourme and T. Duyckaerts].

The author also uses the J. G. Wood and A. J. Bracken paper [J. Math. Phys. 46 (2005), no. 4, article no. 042103], for which he offers a mathematical perspective. The author thoroughly reviews the case of subsets of the plane whose boundary is a conic curve and shows 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, the author shows 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. The author also studies 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.