Revista Matemática Iberoamericana

Time-Frequency Analysis of Sjöstrand's Class

Karlheinz Gröchenig

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We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjöstrand's fundamental results and to far-reaching generalizations.

Article information

Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 703-724.

First available in Project Euclid: 26 October 2006

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Zentralblatt MATH identifier

Primary: 35S05: Pseudodifferential operators 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]

pseudodifferential operators exotic symbols Wigner distribution Gabor frame short-time Fourier transform spectral invariance almost diagonalization modulation space Wiener's Lemma


Gröchenig, Karlheinz. Time-Frequency Analysis of Sjöstrand's Class. Rev. Mat. Iberoamericana 22 (2006), no. 2, 703--724.

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  • Baskakov, A. G.: Wiener's theorem and asymptotic estimates for elements of inverse matrices. Funktsional. Anal. i Prilozhen. 24 (1990), 64-65.
  • Baskakov, A. G.: Asymptotic estimates for elements of matrices of inverse operators and harmonic analysis. Sibirsk. Mat. Zh. 38 (1997), 14-28.
  • Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44 (1977), no. 1, 45-57.
  • Benedetto, J. J., Heil, C. and Walnut, D. F.: Differentiation and the Balian-Low theorem. J. Fourier Anal. Appl. 1 (1995), no. 4, 355-402.
  • Benyi, A., Gröchenig, K., Heil, C. and Okoudjou, K.: Modulation spaces and a class of bounded multilinear pseudodifferential operators. J. Operator Theory 54 (2005), no. 2, 387-399.
  • Björk, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6 (1996), 351-407.
  • Boggiatto, P., Cordero, E. and Gröchenig, K.: Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations Operator Theory 48 (2004), no. 4, 427-442.
  • Bony J.-M. and Lerner, N.: Quantification asymptotique et microlocalisations d'ordre supérieur. I. Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 3, 377-433.
  • Boulkhemair, A.: Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math. Res. Lett. 4 (1997), no. 1, 53-67.
  • Boulkhemair, A.: $L\sp 2$ estimates for Weyl quantization. J. Funct. Anal. 165 (1999), no. 1, 173-204.
  • Cordero, E. and Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205 (2003), no. 1, 107-131.
  • Czaja, W. and Rzeszotnik, Z.: Pseudodifferential operators and Gabor frames: spectral asymptotics. Math. Nachr. 233/234 (2002), 77-88.
  • Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36 (1990), no. 5, 961-1005.
  • Feichtinger, H. G.: On a new Segal algebra. Monatsh. Math. 92 (1981), no. 4, 269-289.
  • Feichtinger, H. G.: Modulation spaces on locally compact abelian groups. Technical report, University of Vienna, 1983.
  • Feichtinger, H. G.: Modulation spaces on locally compact abelian groups. In Proceedings of International Conference on Wavelets and Applications 2002, 99-140. Allied Publishers, Chennai, India, 2003. Updated version of a technical report, University of Vienna, 1983.
  • Feichtinger, H. G. and Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146 (1997), 464-495.
  • Folland, G. B.: Harmonic analysis in phase space. Annals of Mathematics Studies 122. Princeton University Press, Princeton, NJ, 1989.
  • Fornasier, M. and Gröchenig, K.: Intrinsic localization of frames. Constr. Approx. 22 (2005), no. 3, 395-415.
  • Gel'fand, I., Raikov, D. and Shilov, G.: Commutative normed rings. Chelsea Publishing Co., New York, 1964.
  • Gröchenig, K.: Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, 2001.
  • Gröchenig, K.: Composition and spectral invariance of pseudodifferential operators on modulation spaces. To appear in J. Anal. Math. (2006).
  • Gröchenig, K. and Heil, C.: Modulation spaces and pseudodifferential operators. Integral Equations Operator Theory 34 (1999), no. 4, 439-457.
  • Gröchenig, K. and Heil, C.: Modulation spaces as symbol classes for pseudodifferential operators. In Proceedings of International Conference on Wavelets and Applications 2002, 151-170. Allied Publishers, Chennai, India, 2003.
  • Gröchenig, K. and Heil, C.: Counterexamples for boundedness of pseudodifferential operators. Osaka J. Math 41 (2004), 681-691.
  • Gröchenig, K. and Leinert, M.: Wiener's lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc. 17 (2004), 1-18.
  • Heil, C., Ramanathan, J. and Topiwala, P.: Singular values of compact pseudodifferential operators. J. Funct. Anal. 150 (1997), 426-452.
  • Hérau, F.: Melin-Hörmander inequality in a Wiener type pseudo-differential algebra. Ark. Mat. 39 (2001), no. 2, 311-338.
  • Hörmander, L.: The analysis of linear partial differential operators. III. Grundlehren der Mathematischen Wissenschaften 274. Springer-Verlag, Berlin, 1994.
  • Janssen, A. J. E. M.: Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1 (1995), no. 4, 403-436.
  • Labate, D.: Pseudodifferential operators on modulation spaces. J. Math. Anal. Appl. 262 (2001), no. 1, 242-255.
  • Labate, D.: Time-frequency analysis of pseudodifferential operators. Monatsh. Math. 133 (2001), no. 2, 143-156.
  • Meyer, Y.: Ondelettes et opérateurs. II. Opérateurs de Calderón-Zygmund. Actualités Mathématiques. Hermann, Paris, 1990.
  • Pilipović, S. and Teofanov, N.: Pseudodifferential operators on ultra-modulation spaces. J. Funct. Anal. 208 (2004), no. 1, 194-228.
  • Rochberg, R. and Tachizawa, K.: Pseudodifferential operators, Gabor frames and local trigonometric bases. In Gabor analysis and algorithms, 171-192. Appl. Numer. Harmon. Anal.. Birkhäuser, Boston, 1998.
  • Rudin, W.: Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, 1973.
  • Shubin, M. A.: Pseudodifferential operators and spectral theory. Translated from the 1978 Russian original by Stig I. Andersson. Springer-Verlag, Berlin, second edition, 2001.
  • Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1 (1994), no. 2, 185-192.
  • Sjöstrand, J.: Wiener type algebras of pseudodifferential operators. In Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV. École Polytech., Palaiseau, 1995.
  • Strohmer, T.: On the role of the Heisenberg group in wireless communication. Technical report, 2004.
  • Tachizawa, K.: The boundedness of pseudodifferential operators on modulation spaces. Math. Nachr. 168 (1994), 263-277.
  • Toft, J.: Subalgebras to a Wiener type algebra of pseudo-differential operators. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1347-1383.
  • Toft, J.: Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus. Bull. Sci. Math. 126 (2002), no. 2, 115-142.
  • Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal. 207 (2004), no. 2, 399-429.
  • Walnut, D. F.: Lattice size estimates for Gabor decompositions. Monatsh. Math. 115 (1993), no. 3, 245-256.