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Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

  • Thomas Hangelbroek
  • Francis J. Narcowich
  • Christian Rieger
  • Joseph D. Ward
Chapter

Abstract

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given in Hangelbroek et al. (Math Comput, 2017, in press) for restricted surface splines on \({\mathbb {R}}^d\). The kernels for which the theory applies includes the Sobolev-Matérn kernels for closed, compact, connected, C Riemannian manifolds.

Notes

Acknowledgements

Thomas Hangelbroek is supported by grant DMS-1413726 from the National Science Foundation. Francis Narcowich and Joseph Ward are supported by grant DMS-1514789 from the National Science Foundation. Christian Rieger is supported by (SFB) 1060 of the Deutsche Forschungsgemeinschaft.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Thomas Hangelbroek
    • 1
  • Francis J. Narcowich
    • 2
  • Christian Rieger
    • 3
  • Joseph D. Ward
    • 2
  1. 1.Department of MathematicsUniversity of Hawaii – ManoaHonoluluUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.Institut für Numerische SimulationUniversität BonnBonnGermany

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