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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Resolution of the wavefront set using continuous shearlets
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by Gitta Kutyniok and Demetrio Labate PDF
Trans. Amer. Math. Soc. 361 (2009), 2719-2754 Request permission

Abstract:

It is known that the Continuous Wavelet Transform of a distribution $f$ decays rapidly near the points where $f$ is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of $f$. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of $f$ and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by $\mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle$, where the analyzing elements $\psi _{ast}$ are dilated and translated copies of a single generating function $\psi$. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements $\{\psi _{ast}\}$ form a system of smooth functions at continuous scales $a>0$, locations $t \in \mathbb {R}^2$, and oriented along lines of slope $s \in \mathbb {R}$ in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution $f$.
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Additional Information
  • Gitta Kutyniok
  • Affiliation: Department of Statistics, Stanford University, Stanford, California 94305
  • Email: kutyniok@stanford.edu
  • Demetrio Labate
  • Affiliation: Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, North Carolina 27695
  • Email: dlabate@unity.ncsu.edu
  • Received by editor(s): April 24, 2006
  • Received by editor(s) in revised form: November 1, 2007
  • Published electronically: October 24, 2008
  • Additional Notes: The first author acknowledges support from Deutsche Forschungsgemeinschaft (DFG), Grant KU 1446/5-1
    The second author acknowledges support from NSF Grant DMS 0604561
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 2719-2754
  • MSC (2000): Primary 42C15; Secondary 42C40
  • DOI: https://doi.org/10.1090/S0002-9947-08-04700-4
  • MathSciNet review: 2471937