NDLI: Parabolic Molecules

Content Provider	Springer Nature Link
Author	Kutyniok, Gitta Grohs, Philipp
Copyright Year	2013
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.
Ending Page	337
Page Count	39
Starting Page	299
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	2
Volume Number	14
Language	English
Publisher	Springer US
Publisher Date	2013-09-17
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Economics general Parabolic scaling Shearlets Linear and Multilinear Algebras, Matrix Theory Smoothness spaces Curvelets Sparsity equivalence Numerical Analysis Computer Science Applications of Mathematics Math Applications in Computer Science Nonlinear approximation Rate of convergence, degree of approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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