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On Integrals Over a Convex Set of the Wigner Distribution

Abstract

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.

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Fig. 1

Notes

  1. 1.

    On page 2178 of [3], Flandrin writes “it is conjectured that the result (1.1.1) is true for any convex domainC”, a quite mild commitment for the validity of (1.1.1), although that statement was referred to later on as Flandrin’s conjecture in the literature.

  2. 2.

    For \(f\in {\mathscr {S}}({\mathbb {R}}^{N})\), we define its Fourier transform by \( {\hat{f}}(\xi )=\int _{{\mathbb {R}}^{N}} e^{-2i\pi x\cdot \xi } f(x) dx \) and we obtain the inversion formula \( f(x)=\int _{{\mathbb {R}}^{N}} e^{2i\pi x\cdot \xi } {\hat{f}}(\xi ) d\xi \). Both formulas can be extended to tempered distributions.

  3. 3.

    Note that, for \(T_{1}, T_{2}\) distributions on the real line, there is no difficulty at multiplying \(T_{1}(x+y)\) by \(T_{2}(x-y)\): in fact we may define

    $$\begin{aligned} \langle T_{1}(x+y)T_{2}(x-y),\phi (x,y)\rangle _{{\mathscr {D}}'({\mathbb {R}}^{2}),{\mathscr {D}}({\mathbb {R}}^{2})}=\frac{1}{2} \langle T_{1}(x_{1})\otimes T_{2}(x_{2}),\phi \left( \frac{x_{1}+x_{2}}{2},\frac{x_{1}-x_{2}}{2}\right) \rangle _{{\mathscr {D}}'({\mathbb {R}}^{2}),{\mathscr {D}}({\mathbb {R}}^{2})}. \end{aligned}$$

    It is also a general consequence of the location of the wave-front-set of \(T_{1}(x+y)\) (a subset of the conormal bundle of the second diagonal \(x+y=0\)) and of \(T_{2}(x-y)\) (a subset of the conormal bundle of the diagonal \(x-y=0\)): we have

    $$\begin{aligned} \{(x,-x;\xi ,\xi )\in {\mathbb {R}}^{2}\times {\mathbb {R}}^{2} \}\cap \{(x,x;\xi ,-\xi )\in {\mathbb {R}}^{2}\times {\mathbb {R}}^{2} \}=\{(0,0;0,0)\}. \end{aligned}$$
  4. 4.

    For \(t>-1\), we have \(\psi (t)=\int _{0}^{1}\frac{d\theta }{1+\theta t} \) and \(\frac{d}{d t}\{(1+\theta t)^{-1}\}=-(1+\theta t)^{-2} \theta \le 0\).

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Correspondence to Nicolas Lerner.

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Delourme, B., Duyckaerts, T. & Lerner, N. On Integrals Over a Convex Set of the Wigner Distribution. J Fourier Anal Appl 26, 6 (2020). https://doi.org/10.1007/s00041-019-09722-9

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Keywords

  • Wigner distribution
  • Convexity
  • Weyl quantization

Mathematics Subject Classification

  • 35P05
  • 81S30
  • 81Q10