Abstract
We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.
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Notes
- 1.
That is,
$$\displaystyle \begin{aligned} M^1(\mathbb{R}^{2d})\simeq M^1 (\mathbb{R}^d) \widehat{\otimes} M^1 (\mathbb{R}^d),\end{aligned}$$where the symbol \(\widehat {\otimes }\) denotes the projective tensor product (see [35] for the details).
- 2.
Beware that \((A^{\#})_{ij}\neq A^{\#}_{ij} = (A_{ij}^{\top })^{-1}\), i, j = 1, 2.
- 3.
Let \(Qf:\mathbb {R}_{\left (x,\omega \right )}^{2d}\rightarrow \mathbb {C}\) be a time-frequency distribution associated with the signal \(f:\mathbb {R}_{t}^{d}\rightarrow \mathbb {C}\) in a suitable function space. Recall that Q is said to satisfy the strong support property if
$$\displaystyle \begin{aligned} f\left(x\right)=0\Leftrightarrow Qf\left(x,\omega\right)=0, \quad \forall \omega \in \mathbb{R}^d, \qquad \hat{f}\left(\omega\right)=0\Leftrightarrow Qf\left(x,\omega\right)=0, \qquad \forall x\in\mathbb{R}^{d}.\end{aligned} $$ - 4.
With the notation of the previous footnote, we say that Q satisfies the weak support property if, for any signal f:
$$\displaystyle \begin{aligned} \pi_{x}\left(\mathrm{supp}Qf\right)\subset\mathcal{C}\left(\mathrm{supp}f\right), \qquad \pi_{\omega}\left(\mathrm{supp}Qf\right)\subset\mathcal{C}\left(\mathrm{supp}\hat{f}\right), \end{aligned}$$where \(\pi _{x}:\mathbb {R}_{\left (x,\omega \right )}^{2d}\rightarrow \mathbb {R}_{x}^{d}\) and \(\pi _{\omega }:\mathbb {R}_{\left (x,\omega \right )}^{2d}\rightarrow \mathbb {R}_{\omega }^{d}\) are the projections onto the first and second factors (\(\mathbb {R}_{\left (x,\omega \right )}^{2d}\simeq \mathbb {R}_{x}^{d}\times \mathbb {R}_{\omega }^{d}\)) and \(\mathcal {C}\left (E\right )\) is the closed convex hull of \(E\subset \mathbb {R}^{d}\).
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Acknowledgements
E. Cordero and S. I. Trapasso are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). K. Gröchenig acknowledges support from the Austrian Science Fund FWF, project P31887-N32.
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Bayer, D., Cordero, E., Gröchenig, K., Trapasso, S.I. (2020). Linear Perturbations of the Wigner Transform and the Weyl Quantization. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_5
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