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Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces

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.

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Acknowledgments

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

Funding

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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Correspondence to Philipp Petersen.

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Communicated by: Yang Wang

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Petersen, P., Raslan, M. Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces. Adv Comput Math 45, 1581–1606 (2019). https://doi.org/10.1007/s10444-019-09679-9

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Keywords

  • Shearlets
  • Wavelets
  • Sobolev spaces
  • Approximation properties

Mathematics Subject Classification (2010)

  • 42C40
  • 65M60
  • 41A25
  • 65T99
  • 94A12