An inverse theorem for compact Lipschitz regions in $\mathbb {R}^d$ using localized kernel bases
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- by T. Hangelbroek, F. J. Narcowich, C. Rieger and J. D. Ward PDF
- Math. Comp. 87 (2018), 1949-1989 Request permission
Abstract:
While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction recently presented by Fuselier, Hangelbroek and Narcowich [Localized bases for kernel spaces on the unit sphere, SIAM J. Numer. Anal. 51 (2013), no. 5, 2358-2562].References
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Additional Information
- T. Hangelbroek
- Affiliation: Department of Mathematics, University of Hawai\kern.05em‘\kern.05emi – Mānoa, 2565 McCarthy Mall, Honolulu, Hawaii
- Email: hangelbr@math.hawaii.edu
- F. J. Narcowich
- Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
- MR Author ID: 129435
- Email: fnarc@math.tamu.edu
- C. Rieger
- Affiliation: Institut für Numerische Simulation, Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany
- Email: rieger@ins.uni-bonn.de
- J. D. Ward
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 180590
- Email: jward@math.tamu.edu
- Received by editor(s): August 25, 2015
- Received by editor(s) in revised form: May 19, 2016, October 14, 2016, and December 28, 2016
- Published electronically: October 17, 2017
- Additional Notes: The first author’s research was supported by grant DMS-1413726 from the National Science Foundation.
The second author’s research was supported by grant DMS-1514789 from the National Science Foundation
The third author’s research was supported by Collaborative Research Centre (SFB) 1060: The Mathematics of Emergent Effects, of the Deutsche Forschungsgemeinschaft
The fourth author’s research was supported by grant DMS-1514789 from the National Science Foundation. - © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1949-1989
- MSC (2010): Primary 41A17, 41A27, 41A63
- DOI: https://doi.org/10.1090/mcom/3256
- MathSciNet review: 3787398