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

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • Thomas Hangelbroek

    (University of Hawaii – Manoa, Department of Mathematics)

  • Francis J. Narcowich

    (Texas A&M University, Department of Mathematics)

  • Christian Rieger

    (Universität Bonn, Institut für Numerische Simulation)

  • Joseph D. Ward

    (Texas A&M University, Department of Mathematics)

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 ℝ d $${\mathbb {R}}^d$$ . The kernels for which the theory applies includes the Sobolev-Matérn kernels for closed, compact, connected, C ∞ Riemannian manifolds.

Suggested Citation

  • Thomas Hangelbroek & Francis J. Narcowich & Christian Rieger & Joseph D. Ward, 2018. "Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 517-543, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_24
    DOI: 10.1007/978-3-319-72456-0_24
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