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Parabolic Molecules: Curvelets, Shearlets, and Beyond

by P. Grohs and S. Keiper and G. Kutyniok and M. Schaefer

(Report number 2013-47)

Abstract
Anisotropic representation systems such as curvelets and s hearlets have had a significant impact on applied mathematics in the last de cade. The main reason for their success is their superior ability to optimally res olve anisotropic structures such as singularities concentrated on lower dimensional em bedded manifolds, for instance, edges in images or shock fronts in solutions of tra nsport dominated equa- tions. By now, a large variety of such anisotropic systems ha s been introduced, for instance, second generation curvelets, bandlimited shear lets, and compactly sup- ported shearlets, all based on a parabolic dilation operati on. These systems share similar approximation properties, which is usually proven on a case-by-case basis for each different construction. The novel concept of parab olic molecules, which was recently introduced by two of the authors, allows for a un ified framework en- compassing all known anisotropic frame constructions base d on parabolic scaling. The main result essentially states that all such systems sha re similar approximation properties. One main consequence is that at once all the desi rable approximation properties of one system within this framework can be deduce d for virtually any other system based on parabolic scaling. The present paper m otivates and surveys recent results in this direction.

Keywords: Curvelets, Nonlinear Approximation, Parabolic Scaling, Shearlets, Smoothness Spaces, Sparsity Equivalence

@Techreport{GKKS13_547,
  author = {P. Grohs and S. Keiper and G. Kutyniok and M. Schaefer},
  title = {Parabolic Molecules: Curvelets, Shearlets, and Beyond},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-47.pdf },
  year = {2013}
}

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