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Foundations of Computational Mathematics

, Volume 14, Issue 2, pp 299–337 | Cite as

Parabolic Molecules

  • Philipp Grohs
  • Gitta Kutyniok
Article

Abstract

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities.

In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.

Keywords

Curvelets Nonlinear approximation Parabolic scaling Shearlets Smoothness spaces Sparsity equivalence 

Mathematics Subject Classification (2000)

41AXX 41A25 53B 22E 

Notes

Acknowledgements

G. Kutyniok would like to thank Wolfgang Dahmen, David Donoho, Wang-Q Lim, and Pencho Petrushev for enlightening discussions on this and related topics. She acknowledges support by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. The research of P. Grohs was in part funded by the European Research Council under Grant ERC Project STAHDPDE No. 247277.

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Copyright information

© SFoCM 2013

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsETH ZürichZurichSwitzerland
  2. 2.Technische Universität BerlinBerlinGermany

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