Hudson's theorem for \(\tau\)-Wigner transforms (Q2865909)

The Wigner transform NEWLINE\[NEWLINEWig(f)(x,\omega)=\int_{\mathbb{R}^{d}}e^{-2\pi it\omega}f(x+\frac{t}{2}) \overline{g(x-\frac{t}{2})}dt,\quad f\in L^{2}(\mathbb{R}^{d})NEWLINE\]NEWLINE was proposed in 1932 by \textit{E. Wigner} [Phys. Rev., II. Ser. 40, 749--759 (1932; Zbl 0004.38201)]. For \(\tau \in \left[0,1\right]\) and \(f,g\in S(\mathbb{R}^{d})\), the \(\tau\)-Wigner transform is defined as NEWLINE\[NEWLINEWig_{\tau}(f,g)(x,\omega)=\int_{\mathbb{R}^{d}}e^{-2\pi it\omega}f(x+\tau t)\overline{g(x-(1-\tau)t)}dt.NEWLINE\]NEWLINE By a standard density argument, the domain of the \(\tau\)-Wigner transform can be extended to more general spaces, for example the Lebesgue spaces. Also the \(\tau\)-Wigner transform makes sense for \(f,g\in S^{^{\prime }}(\mathbb{R}^{d})\) defining a continuous map: NEWLINE\[NEWLINEWig_{\tau}:S'(\mathbb{R}^{d})\times S'(\mathbb{R}^{d})\rightarrow S'(\mathbb{R}^{2d}).NEWLINE\]NEWLINE It is a natural generalization of the Wigner transform.NEWLINENEWLINEIn this paper the authors prove the two main theorems, Theorem \(2.1\) and Theorem \(2.3\), of positivity of \(Wig_{\tau}(f,g)\) in the settings of square integrable functions and tempered distributions, respectively.NEWLINENEWLINETheorem 2.1. Let us suppose that \(f,g\in L^{2}(\mathbb{R}^{d})\). Then, for every \(\tau \in \left[ 0,1\right] ,Wig_{\tau}(f,g)\in L^{2}(\mathbb{R}^{2d})\), and moreover:NEWLINENEWLINE(i) for every \(\tau \in (0,1),\tau \neq \frac{1}{2},\) we have that \(Wig_{\tau}(f,g)(x,\omega)>0\) almost everywhere (a.e.) in \(\mathbb{R}^{2d}\) if and only if there exist a positive definite matrix \(A\in GL(d,\mathbb{R})\), two vectors \(\alpha ,\beta \in\mathbb{R}^{d}\) and constants \(c,d,a\in\mathbb{R}\) such that NEWLINE\[NEWLINE f(t)=e^{-t^{tr}At+\alpha t+i\beta t+c+i\alpha ,} NEWLINE\]NEWLINE NEWLINE\[NEWLINE g(t)=e^{-(\frac{\tau }{(1-\tau)})t^{tr}At+(\frac{\tau }{(1-\tau)})\alpha t+i\beta t+d+i\alpha ,} NEWLINE\]NEWLINE where as usual NEWLINE\[NEWLINEt^{tr}At=(t_{1}\dots t_{d})\left( \begin{matrix} a_{ 11 } & \dots & a_{ 1d} \\ \vdots & \ddots & \vdots \\ a_{d1} & \dots & a_{dd}\end{matrix}\right) \left(\begin{matrix} t_{1} \\ \vdots\\ t_{d}\end{matrix}\right);NEWLINE\]NEWLINE (ii) (HudsonTheorem) For \(t=\frac{1}{2},\) we have \(Wig_{\frac{1}{2}}(f,g) (x,\omega) =Wig(f,g) (x,\omega) >0\) a.e. in \(\mathbb{R}^{2d}\) if and only if there exist a complex \(d\times d\) matrix \(A\) with positive-definite real part and a vector \(\gamma \in\mathbb{C}^{d}\) such that NEWLINE\[NEWLINE f(t) =e^{-t^{tr}At+\gamma t+c},\text{ }g(t) =\lambda f(t) , NEWLINE\]NEWLINE for an arbitrary positive real constant \(\lambda\).NEWLINENEWLINETheorem 2.3. Let us suppose that \(f,g\in S^{^{\prime }}(\mathbb{R}^{d})\). We consider the ``splitting'' of the variable \(t=(t_{\left[ 1 \right] },t_{\left[ 2\right] },t_{\left[ 3\right] }) \) with \(t_{\left[ 1\right] }=(t_{1},\dots ,t_{h}) ,\) \(t_{\left[ 2\right] }=(t_{h+1},\dots ,t_{k}) ,\) \(t_{\left[ 3\right] }=(t_{k+1},\dots ,t_{d}) \) with \(0\leq h\leq k\leq d\) \((\text{in cases }h=0,\text{ }h=k\text{ and }k=d,\text{ we respectively, mean that }t_{ \left[ 1\right] },t_{\left[ 2\right] }\text{ and }t_{\left[ 3\right] }\text{ are empty})\). We then haveNEWLINENEWLINE(i) for \(\tau \in (0,1) ,\tau \neq \frac{1}{2},\) we have that \(Wig_{\tau}(f,g)(x,\omega)\in S'(\mathbb{R}^{2d})\) is a positive distribution if and only if there exist \( c_{f},c_{g},a\in\mathbb{R},\theta ,\sigma \in\mathbb{R}^{d},D\in GL(d, \mathbb{R}) \) and a splitting of variables of the kind described above such that NEWLINE\[NEWLINE f(t) =e^{c_{f}+i\alpha }U_{D}T_{\theta }M_{\sigma }(e^{-t_{t_{\left[ 1\right] }}^{2}}\otimes \delta _{t_{\left[ 2\right] }}\otimes 1_{t_{\left[ 3\right] }}) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE g(t) =e^{c_{f}+i\alpha }U_{D}T_{\theta }M_{\sigma }(e^{-(\frac{\tau }{(1-\tau) }) t_{1}^{2}\otimes \delta _{t_{\left[ 2\right] }}\otimes 1_{t_{\left[ 3\right] }}}) ,NEWLINE\]NEWLINE where \(\delta _{t_{\left[ 2\right] }}\) is the Dirac distribution in the\ \(t_{ \left[ 2\right] }\)-variable and \(1_{t_{\left[ 3\right] }}\) stands for the function identically \(1\) in the \(t_{\left[ 3\right] }\) variable.NEWLINENEWLINE (ii) \(Wig_{\frac{1}{2}}(f,g) =Wig(f,g) \) is a positive distribution if and only if there exist \(c,a\in\mathbb{R},\theta ,\sigma \in \mathbb{R}^{d},D\in GL(d,\mathbb{R}) ,\) \(A\in GL(h,\mathbb{C}) \) with \(RA\) positive definite, and a splitting of variables as before such that NEWLINE\[NEWLINE f(t) =e^{c_{f}+i\alpha }U_{D}T_{\theta }M_{\sigma }(e^{-t_{ \left[ 1\right] }^{tr}At_{\left[ 1\right] }}\otimes \delta _{t_{\left[ 2 \right] }}\otimes 1_{t_{\left[ 3\right] }}) , g(t) =\lambda f(t) , NEWLINE\]NEWLINE for a positive \(g(t) =\lambda\).NEWLINENEWLINEIn section \(3,\) the authors study some properties of the \(\tau \)-Wigner representation and prove some tecnical results that will be used in the proof of the generalized Hudson's Theorem \(2.1\) and Theorem \(2.3.\) In section \(4,\) the authors give the proofs of Theorem \(2.1\) and Theorem \(2.3.\) In section \(5,\) they give an application of the results to Weyl and localization pseudo-differential operators.