Advertisement

Journal of Fourier Analysis and Applications

, Volume 22, Issue 5, pp 997–1058 | Cite as

Resolution of the Wavefront Set Using General Continuous Wavelet Transforms

  • Jonathan Fell
  • Hartmut Führ
  • Felix Voigtlaender
Article

Abstract

We consider the problem of characterizing the wavefront set of a tempered distribution \(u\in \mathcal {S}'(\mathbb {R}^{d})\) in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group \(H\subset \mathrm{GL}(\mathbb {R}^{d})\). In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group H, called microlocal admissibility and (weak) cone approximation property. Essentially, microlocal admissibility sets up a systematic relationship between the scales in a wavelet dilated by \(h\in H\) on one side, and the matrix norm of h on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives rise to single wavelet characterizations. We illustrate the scope of our results by discussing—in any dimension \(d\ge 2\)—the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for \(d=2\)) holds in arbitrary dimensions.

Keywords

Wavefront set Square-integrable group representation Continuous wavelet transform Anisotropic wavelet systems Shearlets 

Mathematics Subject Classification

42C15 42C40 46F12 

Notes

Acknowledgments

We thank the referees for pointing out valuable references. JF was funded by the ESC-Grant 258926 SPALORA (Sparse and Low Rank Recovery). The research of HF and FV was funded partly by the Excellence Initiative of the German federal and state governments, and by DFG, under the Contract FU 402/5-1.

References

  1. 1.
    Alinhac, S., Gérard, P.: Pseudo-differential Operators and the Nash-Moser Theorem. Graduate Studies in Mathematics. American Mathematical Society, Providence (2007)zbMATHGoogle Scholar
  2. 2.
    Bernier, D., Taylor, K.F.: Wavelets from square-integrable representations. SIAM J. Math. Anal. 27(2), 594–608 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Candès, E.J., Donoho, D.L.: Continuous curvelet transform. I. Resolution of the wavefront set. Appl. Comput. Harmon. Anal. 19(2), 162–197 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coriasco, S., Johansson, K., Toft, J.: Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators. J. Differ. Equ. 254(8), 3228–3258 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Czaja, W., King, E.J.: Isotropic shearlet analogs for \(L^2({\mathbb{R}}^k)\) and localization operators. Numer. Funct. Anal. Optim. 33(7–9), 872–905 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27(2), 195–214 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dahlke, S., Steidl, G., Teschke, G.: The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16(3), 340–364 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahlke, S., Steidl, G., Teschke, G.: Multivariate shearlet transform, shearlet coorbit spaces and their structural properties. Appl. Numer. Harmon. Anal. 86, 105–144 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Delort, J.-M.: F.B.I. Transformation. Second Microlocalization and Semilinear Caustics. Lecture Notes in Mathematics, vol. 1522. Springer, Berlin (1992)zbMATHGoogle Scholar
  10. 10.
    Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  11. 11.
    Führ, H.: Wavelet frames and admissibility in higher dimensions. J. Math. Phys. 37(12), 6353–6366 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Führ, H.: Continuous wavelets transforms from semidirect products. Cienc. Mat. (Havana) 18(2), 179–190 (2000)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    Führ, H.: Generalized Calderón conditions and regular orbit spaces. Colloq. Math. 120(1), 103–126 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Führ, H.: Vanishing Moment Conditions for Wavelet Atoms in Higher Dimensions. http://arxiv.org/abs/1208.2196 (2013)
  16. 16.
    Führ, H.: Coorbit spaces and wavelet coefficient decay over general dilation groups. Trans. Am. Math. Soc. 367(10), 7373–7401 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Führ, H., Mayer, M.: Continuous wavelet transforms from semidirect products: cyclic representations and Plancherel measure. J. Fourier Anal. Appl. 8(4), 375–397 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Führ, H., Voigtlaender, F.: Wavelet coorbit spaces viewed as decomposition spaces. J. Funct. Anal. 269(1), 80–154 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grohs, P.: Continuous shearlet frames and resolution of the wavefront set. Monatsh. Math. 164(4), 393–426 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guo, K., Labate, D.: Characterization and analysis of edges using the continuous shearlet transform. SIAM J. Imaging Sci. 2(3), 959–986 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, K., Labate, D.: Characterization of piecewise-smooth surfaces using the 3D continuous shearlet transform. J. Fourier Anal. Appl. 18(3), 488–516 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Holschneider, M.: Wavelets: An Analysis Tool. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1995)zbMATHGoogle Scholar
  23. 23.
    Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Grundlehren der Mathematischen Wissenschaften, vol. 256, 2nd edn. Springer, Berlin (1990)CrossRefGoogle Scholar
  25. 25.
    Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Am. Math. Soc. 361(5), 2719–2754 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Laugesen, R.S., Weaver, N., Weiss, G.L., Wilson, E.N.: A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. 12(1), 89–102 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Moritoh, S.: Wavelet transforms in Euclidean spaces—their relation with wave front sets and Besov, Triebel-Lizorkin spaces. Tohoku Math. J. (2) 47(4), 555–565 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Murenzi, R.: Wavelet Transforms Associated to the n-Dimensional Euclidean Group with Dilations: Signal in More than One Dimension, Wavelets (Marseille, 1987), Inverse Problems and Theoretical Imaging. Springer, Berlin (1989)zbMATHGoogle Scholar
  29. 29.
    Pilipović, S., Vuletić, M.: Characterization of wave front sets by wavelet transforms. Tohoku Math. J. (2) 58(3), 369–391 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1991)Google Scholar
  31. 31.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jonathan Fell
    • 1
  • Hartmut Führ
    • 2
  • Felix Voigtlaender
    • 2
  1. 1.Lehrstuhl C für Mathematik (Analysis)RWTH Aachen UniversityAachenGermany
  2. 2.Lehrstuhl A für MathematikRWTH Aachen UniversityAachenGermany

Personalised recommendations