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Journal of Fourier Analysis and Applications

, Volume 18, Issue 4, pp 661–684 | Cite as

Approximation of Fourier Integral Operators by Gabor Multipliers

  • Elena Cordero
  • Karlheinz GröchenigEmail author
  • Fabio Nicola
Article

Abstract

A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjöstrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis.

Keywords

Fourier integral operators Modulation spaces Short-time Fourier transform Gabor multipliers 

Mathematics Subject Classification (2000)

35S30 47G30 42C15 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments. K.G. was supported in part by the project P22746-N13 of the Austrian Science Foundation (FWF).

References

  1. 1.
    Asada, K., Fujiwara, D.: On some oscillatory integral transformations in \(L^{2}(\textbf{R}^{n})\). Japan J. Math. (N.S.) 4(2), 299–361 (1978). (Reviewer: Gindikin, S.G., 47G05 (35S05)) MathSciNetGoogle Scholar
  2. 2.
    Boulkhemair, A.: Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math. Res. Lett. 4(1), 53–67 (1997) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Candès, E.J., Demanet, L.: The curvelet representation of wave propagators is optimally sparse. Commun. Pure Appl. Math. 58(11), 1472–1528 (2005) zbMATHCrossRefGoogle Scholar
  4. 4.
    Carathéodory, C.: Variationsrechnung und partielle Differentialgleichungen erster Ordnung. Band I. In: E., Hölder (Hrsg.) Theorie der partiellen Differentialgleichungen erster Ordnung. 2. Aufl. Teubner Verlagsgesellschaft, Leipzig (1956) Google Scholar
  5. 5.
    Concetti, F., Toft, J.: Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols. I. Ark. Mat. 47(2), 295–312 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cordero, E., Gröchenig, K.: Time-frequency analysis of localization operators. J. Funct. Anal. 205(1), 107–131 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cordero, E., Nicola, F.: Boundedness of Schrödinger type propagators on modulation spaces. J. Fourier Anal. Appl. 16(3), 311–339 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cordero, E., Nicola, F., Rodino, L.: Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal. 9(1), 1–21 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cordero, E., Nicola, F., Rodino, L.: Sparsity of Gabor representation of Schrödinger propagators. Appl. Comput. Harmon. Anal. 26(3), 357–370 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Commun. Partial Differ. Equ. 3(11), 979–1005 (1978) zbMATHCrossRefGoogle Scholar
  11. 11.
    Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34(4), 605–612 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983, and also in M. Krishna, R. Radha, S. Thangavelu (eds.), Wavelets and Their Applications, Allied Publishers, pp. 99–140 (2003) Google Scholar
  13. 13.
    Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Feichtinger, H.G., Nowak, K.: A first survey of Gabor multipliers. Advances in Gabor analysis. Appl. Numer. Harmon. Anal., pp. 99–128. Birkhäuser, Boston (2003) Google Scholar
  15. 15.
    Guo, K., Labate, D.: Representation of Fourier integral operators using shearlets. J. Fourier Anal. Appl. 14(3), 327–371 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gröchenig, K.: Foundations of time-frequency analysis. In: Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001) Google Scholar
  17. 17.
    Gröchenig, K.: Time-frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoam. 22(2), 703–724 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gröchenig, K.: A pedestrian’s approach to pseudodifferential operators. In: Harmonic analysis and applications. Appl. Numer. Harmon. Anal., pp. 139–169. Birkhäuser, Boston (2006) CrossRefGoogle Scholar
  19. 19.
    Gröchenig, K.: Representation and approximation of pseudodifferential operators by sums of Gabor multipliers. Appl. Anal. 90(3–4), 385–401 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gröchenig, K., Leinert, M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17(1), 10–18 (2004) CrossRefGoogle Scholar
  21. 21.
    Helffer, B.: Théorie spectrale pour des operateurs globalement elliptiques. Société Mathématique de France, Astérisque (1984) zbMATHGoogle Scholar
  22. 22.
    Helffer, B., Robert, D.: Comportement asymptotique precise du spectre d’operateurs globalement elliptiques dans ℝd. Sem. Goulaouic-Meyer-Schwartz 1980–1981, École Polytechnique, Exposé II, 1980 Google Scholar
  23. 23.
    Krantz, S.: The Implicit Function Theorem. History, Theory, and Applications. Birkhäuser, Boston (2002) zbMATHCrossRefGoogle Scholar
  24. 24.
    Lyubarskii, Yu.I.: Frames in the Bargmann space of entire functions. In: Entire and Subharmonic Functions. Adv. Soviet Math., vol. 11, pp. 167–180. Am. Math. Soc., Providence (1992) Google Scholar
  25. 25.
    Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J. Reine Angew. Math. 429, 91–106 (1992) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ruzhansky, M., Sugimoto, M.: Global L 2-boundedness theorems for a class of Fourier integral operators. Commun. Partial Differ. Equ. 31(4–6), 547–569 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1(2), 185–192 (1994) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Smith, H.F.: A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier 48(3), 797–835 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993). With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III zbMATHGoogle Scholar
  30. 30.
    Olivero, A., Torrésani, B., Kronland-Martinet, R.: A new method for Gabor multipliers estimation: application to sound morphing. In: Proceedings EUSIPCO, Aalborg, Danemark, pp. 507–511 (2010) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Elena Cordero
    • 1
  • Karlheinz Gröchenig
    • 2
    Email author
  • Fabio Nicola
    • 3
  1. 1.Department of MathematicsUniversity of TorinoTorinoItaly
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria
  3. 3.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

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