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 FührEmail author
  • Felix Voigtlaender


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.


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

Mathematics Subject Classification

42C15 42C40 46F12 



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.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

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