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Journal of Fourier Analysis and Applications

, Volume 16, Issue 6, pp 943–982 | Cite as

Quantization of Pseudo-differential Operators on the Torus

  • Michael Ruzhansky
  • Ville Turunen
Article

Abstract

Pseudo-differential and Fourier series operators on the torus \({{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n}\) are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L 2 under certain conditions on their phases and amplitudes.

Pseudo-differential operators Torus Fourier series Microlocal analysis Fourier integral operators 

Mathematics Subject Classification (2000)

58J40 35S05 35S30 42B05 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institute of MathematicsHelsinki University of TechnologyHelsinkiFinland

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