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The basic tools

  • André Unterberger
Chapter
  • 376 Downloads
Part of the Pseudo-Differential Operators book series (PDO, volume 13)

Abstract

Given a number Q > 0, the Weyl calculus OpQ with Planck’s constant Q (in the present investigations, Q will be an integer) associates to every distribution \(\mathcal{G} \in \mathcal{S}^{\prime}(\mathbb{R}^{2})\), the Schwartz space of tempered distributions in the plane, the linear operator OpQ(\(\mathcal{G}\)) from \(\mathcal{S}(\mathbb{R})\) to \(\mathcal{S}^{\prime}(\mathbb{R})\) weakly defined by the equation \(({\mathrm{Op}_{Q}}(\mathcal{G}){w}) \ (x) = {\mathrm{Q}}^{-1} \int_{\mathbb{R}^{2}} {\mathcal{G}}\left(\frac{x + y}{2},{\xi}\right) \ {\mathrm{exp}}\left(\frac{2i\pi}{Q}(x - y){\xi}\right) \ {w}(y) \ {dy} \ {d\xi}\).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • André Unterberger
    • 1
  1. 1.Department of MathematicsUniversity of ReimsReimsFrance

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