Resolution of the Wavefront Set Using General Continuous Wavelet Transforms

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• 作者：Jonathan Fell ; Hartmut Führ…
• 关键词：Wavefront set ; Square ; integrable group representation ; Continuous wavelet transform ; Anisotropic wavelet systems ; Shearlets
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：22
• 期：5
• 页码：997-1058
• 全文大小：1,319 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851
• 卷排序：22

文摘

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.