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Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

  • Philipp PetersenEmail author
  • Mones Raslan
Article
  • 21 Downloads

Abstract

In this paper, we study a newly developed hybrid shearlet-wavelet system on bounded domains which yields frames for Hs(Ω) for some \(s\in \mathbb {N}\), \({\Omega } \subset \mathbb {R}^{2}\). We will derive approximation rates with respect to Hs(Ω) norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-wavelet-based algorithm for the solution of an elliptic PDE and analyze its computational complexity and convergence properties.

Keywords

Shearlets Wavelets Sobolev spaces Approximation properties 

Mathematics Subject Classification (2010)

42C40 65M60 41A25 65T99 94A12 

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Notes

Acknowledgments

P. Petersen and M. Raslan thank P. Grohs and G. Kutyniok for valuable discussions.

Funding information

This work received support from the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and the Berlin Mathematical School. P. Petersen is supported by a DFG Research Fellowship “Shearlet-based energy functionals for anisotropic phase-field methods.”

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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