Continuous shearlet frames and resolution of the wavefront set

Abstract

In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are—unlike more traditional transforms like wavelets—able to efficiently handle data with features along edges. The main result in Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009) confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions ψ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution f with respect to the shearlet ψ can resolve the wavefront set of f. We demonstrate that the same result can be verified under much weaker assumptions on ψ, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for \({L^2(\mathbb{R}^2)}\) from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure.

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Correspondence to Philipp Grohs.

Additional information

Communicated by Karlheinz Gröchenig.

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Grohs, P. Continuous shearlet frames and resolution of the wavefront set. Monatsh Math 164, 393–426 (2011). https://doi.org/10.1007/s00605-010-0264-2

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Keywords

  • Wavefront set
  • Continuous frames
  • Shearlets
  • Microlocal analysis

Mathematics Subject Classification (2000)

  • Primary 42C15
  • Secondary 42C40